31th or 31st is correct?

The numerals with endings are merely abbreviations for the words written out as text. When in doubt, write the word out. Thirty-first becomes 31st, eleventh 11th, forty-second 42nd, fiftieth 50th, and so on.

31st, etc. are the correct options. When you use ordinal numbers ending in 1, you use first. The only exception is eleventh, because although it ends in 1 its “name” doesn’t contain the word “one” like 21, 31, etc.

I think the 101st Airborne Division (U.S. Army infantry division trained for air assault) is a defining case for “standard” usage.

Having said that, in light of 6800 written instances of 101th I can’t say the alternative is “wrong”.