OK, some more actual model names...
Basic running shoes: Asics Cumulus, Asics GT2000 (good stability), Saucony Ride, New Balance 880 (up to v. 14). Puma Velocity Nitro (good outsole, but 10mm heel-to-toe drop is on the high side).
More cushioned versions: Asics Nimbus, Saucony Triumph, New Balance 1080.
Lighter weight and bouncy (and surprisingly good discounts around just now): New Balance FuelCell Rebel.
There are TR versions of all the Asics models and the Ride. Running shoes have been getting lighter and lighter and this has meant road shoes tend to have less and less rubber on the outsole and often don't stand up very well to off-road use. Think of TR variants as more versatile road shoes: you wouldn't buy them as a trail shoe, but for off-road running on tracks and easy footpaths they're all you need.
If you're on the heavier side I'd go for shoes with slightly firmer cushioning, because you will tend to compress the squish out of soft shoes more quickly. Of those I've mentioned the Saucony shoes would probably be the best bet, especially the Triumph (generous quantity of not-too-squishy cushioning).
If most of your runs involve some trail and you don't fancy the NB Hierro then look at the Salomon Aero Blaze Grvl (not as soft or flexible as the Hierro).
Proper trail shoes that I know are runnable on tarmac (but only worth considering if you find yourself running more on trails than on tarmac and struggling with mud or grip in loose terrain) would be the Saucony Xodus Ultra (unusually the older version is available at a good discount just now because they've finally put a better outsole on v4 - the older outsole material is hopelessly slippy on wet rock, although the lugs are good) or the ASICS FujiLite - this has much less cushioning and is very flexible, so probably only suitable if you're reasonably light and light on your feet (but usually available at a great price).