Preliminary notes

First of all it makes no sense to install versions of Ubuntu that are no longer supported with updates. The reasoning is discussed at the bottom of this answer.

This answer will concentrate on currently supported versions of Ubuntu and its official derivatives.

If your hardware never connects to the internet and if you will never use software newer than is included on the respective install media, only then might it be prudent to use outdated versions. But who can be sure of that for all eternity?

You don't have to install Ubuntu to see if it works on your hardware. It is always a good idea to boot from live DVD/USB and see if the system runs okay (although slowly) on the given hardware.

Even if it seems not to work, you might be just one boot option away from a working system. See My computer boots to a black screen, what options do I have to fix it? For example the nomodeset option might help.

Currently supported versions and their minimal requirements

The community wiki usually offers an up-to-date list of currently supported versions. Minimal system requirements for Ubuntu can be found there in the tech specs. The Lists of Ubuntu certified hardware might also be of use to you.

Information on the derivatives is scarce but their requirements are less than what is listed here. Generally 32-bit versions take up less memory and tend to be faster on older systems, than their 64-bit counterparts. However, allot of software nowadays assumes 64-bit architecture. If you are unsure, install the 64-bit versions.

Supported versions of Ubuntu

• 20.04 and later Desktop recommended system requirements

• 2GHz dual core processor or better

• 4GB RAM

• 25GB of disk space

• VGA capable of 1024x768 screen resolution

• Either a DVD drive or a USB port for the installation media

• USB flash drive ≥4 GB for Ubuntu installation media

• Internet access is helpful

• 16.04 Desktop recommended system requirements

• 2GHz dual core processor or better

• 2GB RAM

• 25GB of disk space

• Graphics processor and display capable of at least 1024x768

• Either a DVD drive or a USB port for the installation media

• Internet access is helpful

• 16.04 Desktop minimal

In order to run Unity the system needs a more capable graphics adapter.

• Intel Pentium 4 processor or better

• 1GB RAM

• Graphics processor and display capable of at least 1024x768

• 10 GB of disk space

• 17.10 and later Desktop recommended system requirements

• 64-bit 2GHz dual core processor or better

• 4GB RAM

• 25GB of disk space

• Graphics processor and display capable of at least 1024x768

• Either a DVD drive or a USB port for the installation media

• Internet access is helpful

• 17.10 and later Desktop minimal

• 64-bit processor for iso, Intel Pentium 4 processor or better for installation using the Ubuntu minimal CD. The Ubuntu minimal CD (for Ubuntu 18.04 and earlier) supports installation of Ubuntu on computers that have a 32-bit processor.

• 1GB RAM

• Graphics processor and display capable of at least 1024x768

• 10 GB of disk space

Updates will be provided for Ubuntu 18.04 for 5 years until April, 2023. Updates will be provided for Ubuntu 19.10 for 3 months until July, 2020.

• 18.04 and later Ubuntu Server installation

  • Ubuntu Server supports 3 major architectures: Intel x86, AMD64 and ARM.
  • 1 GHz processor
  • 1 GB of system memory (RAM)
  • 2.5 GB of disk space
  • Graphics card and monitor capable of 640x480
  • USB flash drive ≥2 GB for Ubuntu Server installation media

Security and maintenance updates will be provided for Ubuntu Server 20.04 until April, 2025 and Ubuntu Server 22.04 until April, 2027. Extended Security Maintenance (ESM) will be provided for Ubuntu Server 20.04 until April, 2030 and Ubuntu Server 22.04 until April, 2032 as shown in the below screenshot

^{Ubuntu release cycle}

Supported versions of Xubuntu

See Xubuntu help page.

• 16.04-18.10

Minimum systems requirements
To install or try Xubuntu within the Desktop/Live CD, you need 512MB of memory, 700 MHz processor and 7.5GB of free space on your hard disk.

Recommended systems requirements
To get a smooth experience when running multiple applications in parallel on the desktop, it is recommended to have at least 1GB of memory. It is recommended to have at least 20GB of free disk space. This allows new application installations as well as saving your personal data on the hard disk in addition to the core system.

Security and maintenance updates will be provided for Xubuntu 16.04 for 3 years until April, 2019 and for Xubuntu 20.04 for 3 years until April, 2023. Security and maintenance updates will be provided for Xubuntu 19.10 for 9 months until July, 2020. 

• 19.04 and later

Minimum systems requirements
To install or try Xubuntu within the Desktop/Live CD, you need 512MB of memory, 64-bit 700 MHz processor and 7.5GB of free space on your hard disk.

Recommended systems requirements
To get a smooth experience when running multiple applications in parallel on the desktop, it is recommended to have at least 1GB of memory. It is recommended to have at least 20GB of free disk space. This allows new application installations as well as saving your personal data on the hard disk in addition to the core system.

Supported versions of Lubuntu

• 16.04-17.10

Processor (CPU)
The minimum specification for CPU is Pentium 4 or Pentium M or AMD K8. Older processors are too slow and AMD K7 has problems with flash video.

Memory (RAM)
For advanced internet services like Google+, YouTube, Google Docs and Facebook, your computer needs at least 1GB RAM. Install Lubuntu 32-bit for computers with <= 3GB RAM.

For local programs like LibreOffice and simple browsing habits, your computer needs at least 512MB RAM.

• 18.04

Processor (CPU)
The minimum specification for CPU is Pentium 4 or Pentium M or AMD K8. Older processors are too slow and AMD K7 has problems with flash video.

Memory (RAM)
For advanced internet services like Google+, YouTube, Google Docs and Facebook, your computer needs at least 2GB RAM. Install Lubuntu 32-bit for computers with <= 3GB RAM.

For local programs like LibreOffice and simple browsing habits, your computer needs at least 1GMB RAM.

• 19.04 and later

Processor (CPU)
64-bit processor

Memory (RAM)
For advanced internet services like Google+, YouTube, Google Docs and Facebook, your computer needs at least 2GB RAM.

For local programs like LibreOffice and simple browsing habits, your computer needs at least 1GMB RAM.

Lubuntu Alternate ISOs are for low-RAM PCs. Computers with less than 700 MB of RAM are considered low-RAM computers. Instructions are here.

Updates will be provided for Lubuntu 20.04 for 3 years until April, 2023. Updates will be provided for Lubuntu 19.10 for 9 months until April, 2020.

Supported versions of Ubuntu Core

Ubuntu Core, with an image size of 260MB, is the smallest Ubuntu release to date. This makes it ideal both for IoT devices and cloud containers. A new release of Ubuntu Core is released every 2 years on even numbered years (2016, 2018, 2020, etc.).

Processor – 600MHz processor (ARMv7 or greater, or x86)
System Memory – 128MB RAM or greater
Storage – 4GB flash / storage for factory reset and system rollback

• Ubuntu Core 16, 18 and 20

Ubuntu Core 16 is based on Ubuntu 16.04, and is supported for 5 years.

Ubuntu Core 18 is based on Ubuntu 18.04, and is supported for 10 years.

Ubuntu Core 20 is based on Ubuntu 20.04, and is supported for 10 years.

Ubuntu Core 22 (not stable yet, still in beta) is based on Ubuntu 20.04, and is supported for 10 years.

^{RAM Usage of different flavors of 18.04 under baseline conditions  (Click image to enlarge)}

Why you shouldn't use versions when their support has ended

• Security risks: Eventually there will be an exploit that compromises security or system integrity of old Ubuntu versions
• Software incompatibilities: Versions that are no longer supported will have increasing problems with this. Due to the lack of updates one will no longer be able to open the most recent LibreOffice documents or compile programs that need more recent libraries. Hardware drivers of recent devices will not be included in older kernels.
• Decreasing availability of repositories: It might become very difficult to download software that does not already ship with the outdated version. Hosting repositories for very old versions cease to be economically viable at one point.

Lubuntu or Xubuntu (Ubuntu with the Xfce desktop environment). Xubuntu is more "user friendly" -- more graphical tools for settings, better looking and better integrated applications and maybe better support (larger community).

Lubuntu 18.04 and later needs 1 GB RAM, and Xubuntu 22.04 and later needs 1 GB RAM. Xubuntu and Lubuntu 19.04 and later are available for 64-bit architecture only. For advanced internet services like Google+, YouTube, Google Drive, and Facebook, your computer needs at least 1 GB of RAM. It depends on your skills and preferences, you can try both and then choose.

To install and use Lubuntu 22.04, you need an Intel or AMD 64-bit processor with at least 1 GB of memory. You may need more if you have integrated graphics. When you install Lubuntu 22.04, you need at minimum 10.0 GB of free space on your hard disk (minimum) and 20 GB free space (recommended). Lubuntu 22.04 is a Long Term Support release for which security and maintenance updates will be provided for 3 years until April, 2025.

• Lubuntu: https://lubuntu.me
• Xubuntu: https://xubuntu.org

^{Lubuntu 22.04 with the LXQt desktop environment  (Click image to enlarge)}

Xubuntu

Xubuntu is designed to run on lightweight machines.

^{Xubuntu 22.04  (Click image to enlarge)}

Xubuntu is simply Ubuntu bundled with the Xfce Desktop Environment, a desktop environment designed to be fast and lightweight on lower end systems without compromising performance and visual style.

Xubuntu 18.04-20.04 needs at least 512 MB RAM, 700 MHz processor and 7.5 GB of free space on your hard disk (minimum) and 20 GB free space (recommended).

To install and use Xubuntu 22.04, you need an Intel or AMD 64-bit processor with at least 1 GB of memory. You may need more if you have integrated graphics. When you install Xubuntu 22.04, you need at minimum 8.6 GB of free space on your hard disk (minimum) and 20 GB free space (recommended). Xubuntu 22.04 is a Long Term Support release for which security and maintenance updates will be provided for 3 years until April, 2025.

Xubuntu 19.04 and later is available for 64-bit architecture only.

Ubuntu MATE

Ubuntu MATE is a stable, easy-to-use operating system with a lightweight configurable desktop environment. The MATE Desktop has a rich history and is the continuation of the GNOME 2 desktop, which was the default desktop environment on many Linux and Unix operating systems for over a decade. High DPI will be supported by default in Ubuntu MATE 18.04 and later. The Ubuntu MATE iso file can be written to either a DVD or a USB flash drive that has at least 2GB.

Updates will be provided for Ubuntu MATE 20.04 for 3 years until April, 2023. Updates will be provided for Ubuntu MATE 19.10 for 9 months until July, 2020. If you need Long Term Support, it is recommended you use Ubuntu MATE 20.04 instead.

Ubuntu MATE is compatible with the following devices.

64-bit computers:

• more than 4 GB of RAM
• 64-bit capable Intel (Core 2 Duo or better) and AMD processors
• UEFI PCs booting in CSM mode
• modern Intel-based Apple Macs

GPD Pocket – for the following devices:

• GPD Pocket
• GPD Pocket 2

Raspberry Pi – for aarch32 (ARMv7) computers like:

• Raspberry Pi 2
• Raspberry Pi 3 (also supports Raspberry Pi 2)
• Raspberry Pi 4: Ubuntu 19.10 Raspberry Pi 32-bit and 64-bit preinstalled images (raspi3) now support the Raspberry Pi 4 platform out of the box. You can install Ubuntu Server + MATE desktop environment on a Raspberry Pi 4 by following the instructions in this answer.

PowerPC – for hardware like:

• Apple Macintosh G3, G4 and G5
• Apple iBooks and PowerBooks
• IBM OpenPower 7xx Machines

20.04 and later

16.04-18.04

^{Ubuntu MATE  (Click image to enlarge)}

Ubuntu Budgie

Budgie desktop environment uses many GNOME components and offers a minimalist user interface. Ubuntu Budgie 20.04 will be supported for three years, until April 2023. Ubuntu Budgie 19.10 will be supported for 9 months until July 2020.

20.04 and later

• 64-bit capable Intel and AMD processors
• 4GB RAM
• 25GB disk space
• UEFI PCs booting in CSM mode

18.04

Minimum system requirements:

• 1.5GHz processor
• 2GB RAM
• 20GB disk space

Recommended system requirements:

• 64-bit processor
• 4GB RAM
• 25GB disk space

^{Ubuntu Budgie 21.10  (Click image to enlarge)}

I can disagree on the system requirements for the Linux distros.

On a:

• Intel Pentium 4 1.8Ghz
• 1GB DDR RAM
• 64MB graphics card capable of 1024x768

The only Ubuntu flavour which worked good was Lubuntu.

On a:

• AMD Athlon dual-core 3.20Ghz(OC'ed from 2.80Ghz),
• 2GB DDR3 RAM
• AMD HD 3000 IGP with Catalyst installed.

Ubuntu is really slow (I can work properly with Unity 2D), Gnome Shell is also slow (plus for me Gnome Shell is awful in multitasking). KDE worked quite nice and I'm really impressed by it.

About which flavour to choose I can say this:

• What do you want, better Looks or better Performance?
• XFCE(Xubuntu) looks nicer and its environment has more functionality, you can personalize it and change it in any way you want it (at the cost of high memory use but still you need a good graphic card).
• LXDE(Lubuntu) is faster overall but it lacks the looks.

I suggest to try Xubuntu and if it feels slower go for Lubuntu.

You can try Live-CDs to see how they look but you can test performance only after you installed them.

Have to say that even tough Linux system requirements are really low the desktop environment graphic requirements are high in my opinion.

Well your PC isn't that much hopeless. I would suggest you to try installing normal Ubuntu AND Unity 2D desktop environment.

Of course, if you don't like Unity, this won't be a good solution. However, if you will try that, you will get full Ubuntu support.

I've tried Lubuntu before (installed it via Software manager). It's very basic DE and had some problems. It might be good, but I still suggest you to give Unity 2D a try.