I am in a similar boat, though I have twins at 15 months 
I'm not really sure what classes as normal sleep for babies, so I don't know whether they are particularly bad sleepers, but since about 4 months, they've generally gone to bed at 7pm ish, then woken at 11pm/12pm ish, 2/3am ish, 4/5am ish for a feed (bf) and then up for the day between 6am and 7am. They generally wake separately (thank god - like NewandImproved says, I have no chance of sleep if they both wake together).
Recently they've started waking at 1.30/2am and wanting to get up and play. I think it's linked to their developing walking skills as we went through similar with crawling at 8 months ish.
The only way I can cope is to decamp my duvet to the floor of my bedroom and let them go on the floor. They're used to playing in my room (so it's relatively boring, just a few toys) and the room is baby safe. I then get whatever kip I can on the floor. Eventually they start coming for cuddles and pass out somewhere on me/the duvet/on the boob. Once they're both asleep I put them back in their cots and get back on my bed rather than the floor!
Like others, DH is in the spare room. This works for us as I spend a lot of the night feeding/co sleeping/camping on the floor and there is no point in both of us being up.
I don't know what to suggest really, except that you might get at least some kip on the floor! Other than that - the best thing I did was go back to work
I get nearly an hours sleep on the train every morning and every night
I'm more rested than I have been in 15 months!
Good luck, and at least we can all take comfort that we're not the only ones up at stupid o'clock for no reasonable reason 