The version of winetricks that comes with the stock Ubuntu installation is often quite old, even the version available from a more modern PPA can lag a bit. It is a great idea and a very safe one to update winetricks manually, independent of your main Wine installation, this can be done in a few simple steps:

1. Check for the latest version, remove the old version:

First check to see what is the latest version available from upstream with this handy one-liner:

curl --silent --show-error \
https://raw.githubusercontent.com/Winetricks/winetricks/master/src/winetricks --stderr - \
| grep ^WINETRICKS_VERSION | cut -d '=' -f 2

Then if this trumps your own installed version (to see your version run winetricks --version from a command line) remove your currently installed version:

sudo apt-get remove winetricks

2. Install the latest version:

Then download and install the very latest version:

wget  https://raw.githubusercontent.com/Winetricks/winetricks/master/src/winetricks
chmod +x winetricks 
sudo mv -v winetricks /usr/local/bin

As demonstrated on my own system the latest version can then be tested:

andrew@ilium~$ winetricks --version
20210206-next - sha256sum: e5457a7a984969443119a4c0c1a01607fc8bf8adca114f265669402b24b960b8
andrew@ilium~$ 

3. Add some extras & check for newer syntax:

It may also be a good idea to ensure that you have a few required 'helper' applications for winetricks to run, although most of these should have already been installed with your copy of Wine:

sudo apt-get install cabextract p7zip unrar unzip wget zenity

Then check the correct usage, which could have changed from your older version:

andrew@ilium~$ winetricks -h
Usage: /usr/local/bin/winetricks [options] [command|verb|path-to-verb] ...
Executes given verbs.  Each verb installs an application or changes a setting.

Options:
    --country=CC      Set country code to CC and don't detect your IP address
-f,  --force           Don't check whether packages were already installed
    --gui             Show gui diagnostics even when driven by commandline
    --isolate         Install each app or game in its own bottle (WINEPREFIX)
    --self-update     Update this application to the last version
    --update-rollback Rollback the last self update
-k, --keep_isos       Cache isos (allows later installation without disc)
    --no-clean        Don't delete temp directories (useful during debugging)
-q, --unattended      Don't ask any questions, just install automatically
-r, --ddrescue        Retry hard when caching scratched discs
-t  --torify          Run downloads under torify, if available
    --verify          Run (automated) GUI tests for verbs, if available
-v, --verbose         Echo all commands as they are executed
-h, --help            Display this message and exit
-V, --version         Display version and exit

Commands:
list                  list categories
list-all              list all categories and their verbs
apps list             list verbs in category 'applications'
benchmarks list       list verbs in category 'benchmarks'
dlls list             list verbs in category 'dlls'
games list            list verbs in category 'games'
settings list         list verbs in category 'settings'
list-cached           list cached-and-ready-to-install verbs
list-download         list verbs which download automatically
list-manual-download  list verbs which download with some help from the user
list-installed        list already-installed verbs
arch=32|64            create wineprefix with 32 or 64 bit, this option must be
                      given before prefix=foobar and will not work in case of
                      the default wineprefix.
prefix=foobar         select WINEPREFIX=/home/andrew/.local/share/wineprefixes/foobar
annihilate            Delete ALL DATA AND APPLICATIONS INSIDE THIS WINEPREFIX
andrew@ilium~$ 

4. Safely reverse these steps:

If for some reason you wish to return to your older repository version you can simply run the following:

sudo rm /usr/local/bin/winetricks
sudo apt-get install winetricks

And all will be as it was.

Notes:

• WineHQ Wiki: winetricks All the best winetricks info from the WineHQ Wiki.
• Semi-Automation: It is now possible to partially automate the update process by running sudo winetricks --self-update and if not happy with the results roll this back with: sudo winetricks --update-rollback.

Updated 2019 answer

The winetricks versions that are available in Disco and Eoan are able to update themselves [1].
If you happen to be using 19.04 or later, just run:

sudo apt install winetricks
sudo winetricks --self-update

If you're on an older version of Ubuntu, you can fetch the .deb package here:
https://packages.ubuntu.com/eoan/all/winetricks/download