The concept of regular and irregular primes has played an important role in
number theory at least since the time of Kummer.  We extend this concept to
the setting of arbitrary totally real number fields $k_{0}$, using the
values of the zeta function $\zeta_{k_{0}}$ at negative integers as our
``higher Bernoulli numbers''.  Once we have defined $k_{0}$-regular primes
and the index of $k_{0}$-irregularity, we discuss how to compute these
indices when $k_{0}$ is a real quadratic field.  Finally, we present the
results of some preliminary computations, and show that the frequency of
various indices seems to agree with those predicted by a heuristic
argument.