Various simplicial complexes arising in combinatorics have the Cohen Macaulay property. Such complexes are always (homologically) a wedge of spheres of the same dimension (the dimension of the complex) and also locally have this property. Shellability is an important combinatorial property that again implies the complex and all links to be wedge of spheres. There are interesting extensions of these concepts to the case of spheres of different dimension. Shifted complexes are simplicial complexes whose vertices have a total order and with the property that if S is a face and R is a face of the same dimension with smaller vertices (namely there is a bijection a:R->S so that v<=a(v) for every v) then R is a face. Shifted complexes are always wedge of spheres (of different dimensions) and there are interesting operations which associate to arbitrary simplicial complexes, such shifted complexes.
There are also quite a few interesting classes of simplicial complexes arising in combinatorics which are far from being wege of spheres. Take for example the chessbod complex whose faces are the locations on n nonattacking rooks in an n by n+1 chessboard board. for n=2 it is a hexagon for n=3 a torus for n=4 a certain pseudomanifold all links of vertices are tori, etc.
