I love bloopers, and I love Star Trek. Below, I've collected some classic and more recent Trek bloopers. Prepare to board the Starship Enterbloop! (Blooperprise? Enterbloop, for sure.)
The Original Series
Lots of walking into doors, makeup problems, and cracking up. See also: this collection of clips lacking audio.
The Next Generation/Deep Space 9/Voyager
A lot of TNG Season 1 stuff, including some salty language, then a smattering of DS9 and Voyager bits (including the most extended Vulcan death sequence ever, courtesy of Tuvok). Bonus points to the uploader for having the YouTube username "femmebot1701."
Picard's Alphabet Song
Patrick Stewart does a song-and-dance routine to The Alphabet Song, apparently as a tribute to Gene Roddenberry. It appeared as a bonus feature on the Season 5 DVD set, so it's not technically a blooper, but is a pretty unusual clip! Related: this clip, the latter half of which includes some Sesame Street material.
Star Trek XI Gag Reel
From the J.J. Abrams reboot, a special feature from the home video release. Lots of bleeps.
Got More Bloopers?
Post 'em in the comments! Most of the others I could find on YouTube appeared to be derived from the clips above, so you may have to go digging.
See also: The Late Movies: Bloopers A-Plenty, where a few of these videos were seen before.