I know that Rabbeinu Tam holds that you should wait 72 minutes after sunset to end Shabbat. What is the Halachic basis for other opinions? (42 min. etc.)?
The basic problem is, what Halachicaly is sunset. According to Rabbeinu Tam "Biat Shemesh" means the time that the sun no longer lights up the sky, which is considerably later than our sunset. "Tzeit Hacochavim" - the shining of the stars is then later, when the sky is quite black. According to most Rishonim, sundown is when the sun disappears behind the horizon, and Tzeit Hacochavim is when stars can be discerned. Obviously, that is considerably earlier than 72 minutes, but the exact time will depend on which size stars you have to see.