  We study the coregularity of smooth Fano threefolds. We prove that for 100
out of 105 families of smooth Fano threefolds, a general member in the family
has coregularity 0; moreover, for 92 families out of these 100, any member in
the family has coregularity 0; for the remaining 5 families, we obtain some
partial results. In particular, we show that there exist families of smooth
Fano threefolds whose general elements have positive coregularity.
