How do I remove only shopping searches?

You can't remove the Unity Shopping Lens in Ubuntu 13.10 because there's no Shopping lens. Ubuntu 13.10 comes with a feature called Unity Smart Scopes (or "100 scopes") which uses a huge list of scopes to display results in Dash and among these, there are some used for shopping. So if you don't want shopping suggestions in Dash, you must disable these shopping scopes (see below). Unity Dash without shopping suggestions

How to disable Unity Dash plugins (scopes) in Ubuntu 13.10

To disable a Dash plugin (scope) in Ubuntu 13.10 Saucy Salamander:

• open Dash,
• go to the Applications lens (either do it manually using your mouse or use the Super + A keyboard shortcut),
• click "Filter results" on the right and under "Type",
• select "Dash plugins".

All the Dash search plugins (scopes) should be listed.

To disable a plugin, click on it and then click the "Disable" button. Later on you can enable them back in the same way.

Disable the Amazon / shopping suggestions Unity Dash scopes

If you don't want shopping suggestions in Dash, disable the following scopes (by using the instructions above): Amazon, Ebay, Music Store, Popular Tracks Online, Skimlinks, Ubuntu One Music Search and Ubuntu Shop.

To disable all these shopping Dash plugins/scopes from the command line, use the following command

 gsettings set com.canonical.Unity.Lenses disabled-scopes "['more_suggestions-amazon.scope', 'more_suggestions-u1ms.scope', 'more_suggestions-populartracks.scope', 'music-musicstore.scope', 'more_suggestions-ebay.scope', 'more_suggestions-ubuntushop.scope', 'more_suggestions-skimlinks.scope']"

There's now a GUI for doing this.

Open "System Settings" (type that in the Dash). Then go to "Security and Privacy". Then "Search". Then set the "include on-line search results" to off.

Disclaimer: It seems likely that this prevents the system from phoning home with everything typed in to Dash, but I'm not positive of this.

It turns out that Fix Ubuntu has the solution:

#!/bin/bash

# Figure out the version of Ubuntu that you're running
V=`/usr/bin/lsb_release -rs`;

# The privacy problems started with 12.10, so earlier versions should do nothing
if [ $V \< 12.10 ]; then
  echo "Good news! Your version of Ubuntu doesn't invade your privacy.";
else
  # Turn off "Remote Search", so search terms in Dash don't get sent to the internet
  gsettings set com.canonical.Unity.Lenses remote-content-search none;

  # If you're using earlier than 13.10, uninstall unity-lens-shopping
  if [ $V \< 13.10 ]; then
    sudo apt-get remove -y unity-lens-shopping;

  # If you're using a later version, disable remote scopes
  else
    gsettings set com.canonical.Unity.Lenses disabled-scopes \
      "['more_suggestions-amazon.scope', 'more_suggestions-u1ms.scope',
      'more_suggestions-populartracks.scope', 'music-musicstore.scope',
      'more_suggestions-ebay.scope', 'more_suggestions-ubuntushop.scope',
      'more_suggestions-skimlinks.scope']";
  fi;

  # Block connections to Ubuntu's ad server, just in case
  if ! grep -q productsearch.ubuntu.com /etc/hosts; then
    echo -e "\n127.0.0.1 productsearch.ubuntu.com" | sudo tee -a /etc/hosts >/dev/null;
  fi;

  echo "All done. Enjoy your privacy.";
fi

For 13.10, that's gsettings set com.canonical.Unity.Lenses disabled-scopes "['more_suggestions-amazon.scope', 'more_suggestions-u1ms.scope', 'more_suggestions-populartracks.scope', 'music-musicstore.scope', 'more_suggestions-ebay.scope', 'more_suggestions-ubuntushop.scope', 'more_suggestions-skimlinks.scope']"; and edit /etc/hosts to redirect requests for productsearch.ubuntu.com to 127.0.0.1 (localhost)

I am assuming you want to remove some other lenses too. So first get an overview of what is actually installed:

Open a terminal: CTRLALTT and type

 apt-cache policy "unity-lens-*"|grep -B1 Installed

This will give you a list of the form:

unity-lens-video:
  Installed: (none)
--
unity-lens-shopping:
  Installed: (none)
--
unity-lens-friends:
  Installed: 0.1.1bzr13.04.12-0ubuntu1
--
....

Now you can decide what you want to remove and do so with e.g.

sudo apt-get remove unity-lens-friends

until the desired state of your system is reached.

Whenever you are in doubt what a lens will do you can get the needed information with apt-cache show, e.g.

apt-cache show unity-lens-files

A little way down you will see:

Description-en: File lens for unity
 This package contains the "file" lens which can be used
 inside Unity to browse your files.