  The philosophy of the article is that the desingularization invariant
together with natural geometric information can be used to compute local normal
forms of singularities. The idea is used in two related problems: (1) We give a
proof of resolution of singularities of a variety or a divisor, except for
simple normal crossings (i.e., which avoids blowing up simple normal crossings,
and ends up with a variety or a divisor having only simple normal crossings
singularities). (2) For more general normal crossings (in a local analytic or
formal sense), such a result does not hold. We find the smallest class of
singularities (in low dimension or low codimension) with which we necessarily
end up if we avoid blowing up normal crossings singularities. Several of the
questions studied were raised by Kollar.
