  We establish {\L}ojasiewicz inequalities for a class of cylindrical
self-shrinkers for the mean curvature flow, which includes round cylinders and
cylinders over Abresch-Langer curves, in any codimension. We deduce the
uniqueness of blowups at singularities modelled on this class of cylinders, and
that any such cylinder is isolated in the space of self-shrinkers. The
Abresch-Langer case answers a conjecture of Colding-Minicozzi. Our proof uses
direct perturbative analysis of the shrinker mean curvature, so it is new even
for round cylinders.
