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.
By far the best bagels in London, they are open 24 hours and being just down the road from my flat a bit too local. It is run by a man called Mr Sammy, who doesn't muck about with portion size. It is well worth a visit especially for their smoked salmon and cream cheese speciality.
It's really close to my house and one of my favourite pubs in London! (I even went before it was close to where I lived!) It is London's first cooperatively owned pubs and has lovely events and workshops, exercise classes and great parties, and good beer! It's lovely in the beer garden in the hot summer and equally great in the winter, next to a real fire.
Set amongst one of my favourite housing estates in London – the Brunswick, Renoir cinema shows art-house films in it's two small but comfortable screens. I come here at least twice a month, and always on my own.