I have used Adidas trainers before, they are ok but not great. If i knew the names off the top of my head i would name but i cant. They are low in cost though, so if your getting through a lot of trainers in a year, then they are good purchase for that reason.
I had some NB, i think they were the 1036, they were really nice, comfy shoes. I did really enjoy these ones, i rate them as good as anything ive had.
Ive had a few different pairs of asics and ive always found these a pleasure to run in, i think the last pair i ran in were the 2120 or something like that. I just wish they didnt alwyas come with such high price tags!
I did once have a pair of Saucony, but i didn't enjoy running long distances in them at all and quickly moved on from them.