We saw some great museum shows while in LA. I really enjoyed the fact that we could see interesting modern and contemporary work in the museums, as the NYC ones haven't been great about that lately with the exception of the Whitney.
I already mentioned the (now closed) minimalism show at MOCA, which was brilliant. Other shows we saw of note:
Made in Mexico, Rob Voerman, and Tara Donovan at the UCLA Hammer. I especially liked the work by Anton Vidokle and Andrea Fraser's video. It's one of my favorite works by her that I've seen.
Beyond Geometry at LACMA. I agree with Tyler Green that the premise is a bit baffling, but I loved the show. We didn't make it to the show on race and 18th century colonial Mexican painting, but we wanted to. LACMA is free after 5PM all summer so you get a cooler mix of people at the shows than a museum might normally have. I like the open terraces between the buildings there.
If you like Tara Donovan's work, you should also check out Sharon Louden. See here and here.