It is OK to run in training shoes although unless you're running a short distance less than one mile, it's generally not a good idea to wear training shoes for running.
Not only can training shoes be heavier and more cumbersome, but they also don't have sufficient cushioning to absorb the shock of repeatedly pounding your feet into the pavement.
Some of the best training shoes are.
Reebok Nano X2.
Mizuno Wave Rider 25.
Brooks Launch 8.
Ryka Myriad.
Tiem Slipstream.
Adidas Powerlift 5.
Nike Free Metcon 4.
Under Armour UA TriBase Reign 3.