For your own site, you’re your own client, so it’s up to you how much QA work you want to do.
Different browsers still have different capabilities and behaviors, so ensuring your site works OK across the most popular browsers is useful.
Here are global stats, but if you were targeting a specific region of the world, you’d want to check what browsers are popular there ( QQ in China, etc. ). Jeff also pointed out that people do not update phones consistently, so there are a lot of different versions of different browsers in the wild.
Take a look at these reports to understand more.