When I decided to write this post, I did a quick google search to see what is available. It was really embarrassing. Almost all the advice being given for this was from websites that earn a commission if you click on link of the running shoes to buy them. Most of the shoes being recommended would not help you at all if you had bunions.
There is plenty of good information on bunions online, so there is no point litigating what they are and what causes them here, except to say that the only way to get rid of them is with surgery and the best you can do to prevent them getting worse or being so painful is to get the shoe fitting correct.
The running shoes that are best if you have bunions are those that are wider in the forefoot to accommodate the enlarged joint. Your typical running shoe made on a wider last is often not wide enough (though you will read commonly that they are from those who make money selling those shoes). You could try on the wider last models of each of the brands to see how they feel on you. In all honestly, there is only one brand that is any good here and they are the Altra running shoes. They are massively wide in the forefoot compared to other brands and they have the width to accommodate the enlarged joint associated with the bunion. If you have a bunion, then try a pair.
What else could you do for bunions?
Apart from surgery, there are other options. What is important is to keep the joint mobile and flexible with exercises as it does have a tendency to become stiffer. A bunion corrector can also be used to help with that. If you can not get the shoes right, then there are donut shaped pads that can be worn to keep the pressure off the joint.