  In this paper we are concerned with three lattice problems: the lattice
packing problem, the lattice covering problem and the lattice packing-covering
problem. One way to find optimal lattices for these problems is to enumerate
all finitely many, locally optimal lattices. For the lattice packing problem
there are two classical algorithms going back to Minkowski and Voronoi. For the
covering and for the packing-covering problem we propose new algorithms. Here
we give a brief survey about these approaches. We report on some recent
computer based computations where we were able to reproduce and partially
extend the known classification of locally optimal lattices. Furthermore we
found new record breaking covering and packing-covering lattices. We describe
several methods with examples to show that a lattice is a locally optimal
solution to one of the three problems.
