The IMF, in 2011, claimed :
“a 10 percent permanent increase in oil prices reduces oil demand by about 0.7 percent after 20 years.” [1]
So to put things simpler : there is no elasticity in the price oil to speak of. It is almost like this : for all intents and purposes, people will die, rather than stop buying oil (in most of Northern Europe, of course, people will die without oil, ditto for most of Canada, Alaska, Russia, ...) (another way to say this would be : look at why the US/"West" went to war at any point after Vietnam. Decide for yourself why those wars happened, what would be the cause ? Oil price. Even ISIS and Sudan, to a significant extent, are oil wars. Well, it's muslim lunatics, but they wouldn't be a threat to anyone without oil)
As for peak oil, it's happening. Oil production has plateau'd since 2005. Conventional oil peaked in 1998, but horizontal drilling got it to go back up. In 2007 we really started to see a drop of 1-2%, but then the shale revolution happened (massive production spike, supposedly short-lived, in America matching declines in pretty much everywhere else in the world). Of course, the shale revolution skipped Europe. This is distorting the picture in America. In America it really looks like oil production has gone up, but that is not true globally.
Recently, (massive) consumption slowdowns in China have masked a small decline, but that's probably noise. To be honest, it is extremely scary that this has not lead to a significant price decline (talking 5-10% here, we've seen 1-2% at most).
I would also argue that while small declines are indeed noise, there's 2 factors that I feel still deserve some attention.
1) Since 2005, oil has been on a very small, but very (very) steady decline. The amounts are less than the noise, but it's hard looking at the graph and not see a decline.
2) Production has remained somewhat steady, but consumption has definitely shifted. Firstly internally : middle eastern countries' oil consumption has grown on the exponential curve. The same goes for pretty much all significant oil production countries. Since the internal market is subsidized in nearly all cases, this means that effective oil available on the international market (where e.g. Europe has to buy it) has gone down 5-8%.
3) The fact that the US did not have a way to export it's oil is masking fuel prices within the United States. It's one more factor that's making oil prices lower in the US than you'd expect given the rest of the world.
[1] http://www.imf.org/external/pubs/ft/weo/2011/01/pdf/text.pdf