I have a betta in a 1.5 gallon cube that is filtered and heated. He is very happy in it and actually started tail biting when I upgraded him to a 3 gallon, and his tail has improved since I moved him back down to the 1.5 gallon.
I also have a betta in a heated 2 gallon glass "drum" bowl. I have been able to fit a small filter (Tetra Whisper 3i that suction cups to the side) in there, but it's currently being borrowed by a friend of mine.
Working at a pet store, I will say very bare minimum of a tank should be no less than 1.5 gallons since a lot of people in this area will refuse to listen if I tell them 2.5 gallons or higher. Many think it's CRAZY to have a single betta in a 10 gallon.
All my other bettas are in 3 gallon tanks and one lucky guy is in a 10 gallon with some neons and corydoras.