Maybe an obvious choice, but the Barbican is always good. My favourite space in there is the curve gallery, last time I went was for Cory Arcangel's 'Beat the Champ'. I love the architecture, although not many people do. I think it was voted 'London's ugliest building' in 2003.
Love to visit this place at least once every couple of months, can't afford most of the stuff but at least I can take a picture of myself wearing just for a few seconds this beautiful Comme des Garçons Mickey Mouse Hat!