For almost any purpose, Pearson correlation is what you want. The others are experimental.
Spearman would be used if you expect a lot of outliers. Usually it will be almost identical to Pearson.
Taub's k is really useful only when the data consists of a few levels (such a numbers 1, 2, 3, 4, or 5), not a continuously variable value.
Quadrant correlation is never useful, and is there just for fun fun fun in the sun sun sun.