  We prove tight H\"olderian error bounds for all $p$-cones. Surprisingly, the
exponents differ in several ways from those that have been previously
conjectured; moreover, they illuminate $p$-cones as a curious example of a
class of objects that possess properties in 3 dimensions that they do not in 4
or more. Using our error bounds, we analyse least squares problems with
$p$-norm regularization, where our results enable us to compute the
corresponding KL exponents for previously inaccessible values of $p$. Another
application is a (relatively) simple proof that most $p$-cones are neither
self-dual nor homogeneous. Our error bounds are obtained under the framework of
facial residual functions, and we expand it by establishing for general cones
an optimality criterion under which the resulting error bound must be tight.
