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 McKay, right at the beginning of the story, and consequently owe him a meal. But that loss is offset by the far greater gain of, hey, having a meal with Chris McKay.
And another gain is a greater appreciation of the power of theory. The observational case for methane on Mars was pretty good -- good teams, different techniques. The theoretical case against it, though, as brilliantly articulated by my friends Kevin Zahnle and David Catling and their colleague Richard Freedman in this article (pdf), was really persuasive. That insight has broader applicability, as I noted here:
Thomas Huxley, Darwin's ally in the fight to get evolution accepted, spoke warmly of the facility with which ugly facts can kill beautiful theories. But that fatal ability should not hide the fact that well-applied theories, beautiful and otherwise, can play a crucial role in deciding which observations get treated as facts in the first place.