The following telescopes certainly have adaptive optics systems:
Admittedly, some of these links aren't exactly clear, but I think they all confirm the presence of AO systems in those instruments.
It doesn't appear that SALT uses adaptive optics, though there are proposals to implement it.
It should be mentioned that adaptive optics are advancing. At Gemini Observatory (for which I work, but don't represent in any official capacity) we are very proud of a new instrument that just recently saw "first light" that projects an artificial "constellation" of artificial guide stars, allowing for a larger field of correction: .
It may also be worth mentioning that adaptive optics don't necessarily require an artificial guide star, one can use a real star, if one happens to be near the object being studied (among other constraints). My current project is programming a "guide star recommendation" tool for just such situations (fun!).
Many of the ground-based solar telescopes, particularly those that look at subfields (less than the full solar disk) have adaptive optics. I'm probably missing some, but the list currently includes:
In addition to the image quality issues, for large >4 m telescopes, AO is almost an engineering requirement.
To make the dome affordable, you have to make very fast optical systems, which lead to very large plate scales so to build reasonable sized instruments (especially spectrographs) you need to get the star image down to a small physical size, by keeping the angular size of the image small. The only way to do this is to reduce the degree of blurring.
I think most big telescopes have AO now. I found references for Palomar, Lick, Keck and CFH (Canada France Hawaii). Apparently AO was developed during the nineties and was so good that it spread rapidly.