  We deal with the asymptotic enumeration of combinatorial structures on planar
maps. Prominent instances of such problems are the enumeration of spanning
trees, bipartite perfect matchings, and ice models. The notion of orientations
with out-degrees prescribed by a function $\aa:V\to \NN$ unifies many different
combinatorial structures, including the afore mentioned. We call these
orientations $\aa$-orientations. The main focus of this paper are bounds for
the maximum number of $\aa$-orientations that a planar map with $n$ vertices
can have, for different instances of $\aa$. We give examples of triangulations
with $2.37^n$ Schnyder woods, 3-connected planar maps with $3.209^n$ Schnyder
woods and inner triangulations with $2.91^n$ bipolar orientations. These lower
bounds are accompanied by upper bounds of $3.56^n$, $8^n$ and $3.97^n$
respectively. We also show that for any planar map $M$ and any $\alpha$ the
number of $\alpha$-orientations is bounded from above by $3.73^n$ and describe
a family of maps which have at least $2.598^n$ $\alpha$-orientations.
