The 135L is a very sharp lens and is probably considered Canon's sharpest prime.
Perhaps sharpest prime under $5000? The 200/2, 300/2.8, etc., are noticeably sharper.
Where I am puzzled is I hearing more and more (see Neuro comments above) that the 70-200 2.8 II IS is almost as sharp as the 135L wide open. It is more versatile but a bit heavier. I am debating to buy this zoom lens instead. Although for indoor use the 135L might be better due to faster prime.
When shooting both lens at f2.8, the test out there suggest the 135 is much sharper, but from real user, anyone experience with these and can comment?
I wouldn't say the 135/2 is 'much sharper' than the 70-200/2.8 II at f/2.8 - they are pretty close (TDP's ISO 12233 crops show a very slight advantage to the 135/2, photozone's resolution figures give a very slight edge to the 70-200 II away from the center) - note, that's the MkII, which is sharper than any of the other 70-200/2.8 lenses. In real world use, those differences are not going to be visible at all.
The 135L does let in an extra stop of light, but I'd really say it depends on your use for the lens (indoors or out). If you're shooting people under standard conditions (events, etc.), 1/60 s is usually enough to freeze movement. At 135mm, you need about 1/125 s to have a decent keeper rate (assuming FF, on APS-C more like 1/200 s). So, in dim light the 135L will give you the extra stop, with the 70-200/2.8 you have IS to allow you to handhold at 1/60 s. Where the 135L will make the difference in low light is shooting action, where you need all the shutter speed you can get. Also, it makes a difference in the amount of OOF blur for portraits.
So, IMO the 135L is better for indoor sports, dance recitals, etc., and better for portraits - I use mine for those purposes. Otherwise, the 70-200mm f/2.8L IS II is better for general purpose use (assuming you don't mind the weight, and have the budget for it) - and with the great IQ and versatility, the 70-200 II is my second most-used lens.