  We present a set of 26 finite quandles that distinguish (up to reversal and
mirror image) by number of colorings, all of the 2977 prime oriented knots with
up to 12 crossings. We also show that 1058 of these knots can be distinguished
from their mirror images by the number of colorings by quandles from a certain
set of 23 finite quandles. We study the colorings of these 2977 knots by all of
the 431 connected quandles of order at most 35 found by L. Vendramin. Among
other things, we collect information about quandles that have the same number
of colorings for all of the 2977 knots. For example, we prove that if $Q$ is a
simple quandle of prime power order then $Q$ and the dual quandle $Q^*$ of $Q$
have the same number of colorings for all knots and conjecture that this holds
for all Alexander quandles $Q$. We study a knot invariant based on a quandle
homomorphism $f:Q_1\to Q_0$. We also apply the quandle colorings we have
computed to obtain some new results for the bridge index, the Nakanishi index,
the tunnel number, and the unknotting number. In an appendix we discuss various
properties of the quandles in Vendramin's list. Links to the data computed and
various programs in C, GAP and Maple are provided.
