I've lived in two and have a brother that lives in the other
I personally wouldn't live in any of the city propers, but let me give you a rundown of what I think. All three city leaderships suck.
Memphis - if you want more stuff to do, this is your winner. Memphis has the Grizzlies and a top-notch concert venue in the Fedex Forum, a good nightlife district downtown, etc. No state income tax in Tennessee, and some nice suburbs. You are four hours from St. Louis, if you want a little more. Its bigger, so there are more jobs, and more good companies to work for, but theres a lot of competition for those jobs - especially in the young recent college grad arena. Because it is bigger, there are some really dangerous areas.
Birmingham - probably the worse of the three in my opinion, unless you are in healthcare - lowest wages of the three. Schools are absolutely terrible. Hoover schools are good, not great, and you are paying a premium there so you expect more. Good shopping in Mountain Brook/280 corridor. 2 hours from Atlanta is a big, big plus in my opinion.
Jackson - Less populated = less stuff to do, but shopping is comparable to both Memphis and Bham. 3 hours from Memphis, NOLA and the coast. Madison and Flowood are really nice places to raise a family. There aren't as many high paying jobs, but f you can land one, your money goes further than the other two. But if you are young and always want something to do, somewhere to go - then Jackson isn't for you.