Data set is available on Kaggle. Some data also from Wikipedia.
Between 2005 and 2007 the US accepted about 40,000 asylum seekers. By comparison the UK accepted 30,000 and Canada 25,000. However the population of the US is several times larger than the populations of the UK and Canada combined so per-capita the US was not taking in as many people as either the UK or Canada. There are two kinds of asylum granted in the US, affirmative in which the application is accepted and defensive in which the original application is rejected, the case then goes in front of a judge who can decide to accept or reject it.
So since 2007 the number of people being granted affirmative asylum has incresed significantly while those getting defensive asylum has decreased. The origins of these asylum seekers has also changed over the last decade or so.