Or at least very, very little. Observations made with the Tunable Laser Spectrometer on Curiosity (paper|press release) rule out an atmospheric level above 1.3 parts per billion with 95% confidence: the data is fully consistent with no methane at all. As the authors say:
Our result greatly reduces the probability of significant methanogenic microbial activity on Mars and recent methane production by serpentinization or from exogenous sources including meteoritic, interplanetary dust and cometary infall.
There will be some background from meteoritic sources, I assume, at the parts per trillion level (where I imagine they will stay undetectible even if Mars passes through the coma of comet C/2013 A1 (Siding Spring)). But these observations leave the idea of a methane-producing cryptic biosphere with no supporting evidence at all. This would feel like the end, if it didn't feel like things had ended a while back -- see this post and links therein. I rather regret that Mars looks deader still than it did before, but I don't regret having been so excited by the subject back in the day -- it's the nature of the game.
My only real loss is that I made a bet on the subject with Chris ...