These combos are based off of the basic attack mode without holding R1. These will help you get a better combo rating, especially if you do the first and follow with the second. Make sure not to hold R1 button or you will do the Air Hike or the Stinger.
Three basic slashes plus a downward strike, then an upward strike that knocks your enemy into the air like the Air Hike: Press Triangle(3), Towards + Triangle, Triangle(2).
Three basic slashes then three additional horizontal strikes. Press Triangle(3), Away + Triangle, Triangle(3).
Equip any sword and the shotgun. Holding R1 and do the following:
Dante will do a normal slash, then a cartwheel move while firing the shotgun. Press Triangle, Square.
Dante will do two slashes and pass the shotgun under his two arms and across his back, firing the shotgun twice. Press Triangle(2), Square.
Dante will do three slashes and then go into one knee firing the shotgun. Press Triangle(3), Square.