Currently, I'm wearing Asics to run in. I love love love their fit and my feet felt like they were coming home when I put them on. I"ve run and trained for a half marathon in them and I'll be training for my first marathon with them as well. I can't wait until I get my next pair. I'm due for one as mine have too much mileage on them.
Before my Asics, however, I was a die-hard New Balance fan. I have quite narrow feet and benefitted greatly from the NB width sizing.
I would go for either NB or Asics.
As for cross-trainers, eh...well, I'm not a fan. You know: "a jack of all trades and the master of none" etc etc. I don't think it's absolutely necessary, but I tend to keep my NBs at the gym and my Asics at home. It's nice for your feet (and therefore the rest of your body) to get challenged in different ways. Also, if you go for a hard workout, you can alter the cushioning of your shoe. If you wear them before they've "recovered", they won't give you all the support, cushioning, etc that they're supposed to.
ok. I'm off my soapbox now.