We always like going to the Barbican by bike because the cycle parking's so good. Lots of simple, straightforward, secure racks (right), conveniently shaped for a lock on each wheel, not liable to let your bike slide and fall over, and most delightfully, immediately outside the Silk St entrance. Other attractions take note.
They lose a couple of marks for a rather vague, mis-apostrophised web page about it that navigates you by the car parks (no, we don't need the car park, that's why we're on bikes). But when their Monday night brand-new feature films only cost a fiver, we can forgive that.