  We construct local models of Shimura varieties and investigate their
singularities, with special emphasis on wildly ramified cases. More precisely,
with the exception of odd unitary groups in residue characteristic $2$ we
construct local models, show reducedness of their special fiber,
Cohen$-$Macaulayness and in equi-characteristic also (pseudo-)rationality. In
mixed characteristic we conjecture their (pseudo-)rationality. This is based on
the construction of parahoric group schemes over two-dimensional bases for
wildly ramified groups and an analysis of singularities of the attached
Schubert varieties in positive characteristic using perfect geometry.
