Erase certain messages in Messages on OS X

Solution 1:

Deleting messages on an iOS device doesn't delete them from any other device.

• You can remove message threads by clicking the × on the right of the contact.

  

• You can remove individual messages by right-clicking the message and selecting Delete.

  

Solution 2:

There's an important detail missing in this answer, for those of you/us who are "click challenged". The menu shown above (Forward...,Copy, Delete...) only shows up if you are clicking in the correct part of the message. If you click over any of the words, you'll get a menu that looks like this

If you instead click on an area of the message that has NO words, you'll get the menu shown in grgarside's post above.

Happy deleting!