There are, up to homeomorphism, three distinct topologies on the two-point set $X=\{x,y\}$. Of these, the indiscrete space and the Sierpinski space are contractible, while the discrete space is not even path connected. Thus these spaces represent only two distinct homotopy types.
On the other hand, the three-point set $Y=\{x,y,z\}$ admits nine distinct topologies.
How many distinct homotopy types do these nine topologies represent?
I'm sure the answer to this must be in Barmak's thesis or May's notes, but I couldn't find it in either. It should be straightforward to answer this if you understand the correct machinery (as of yet I do not).
Another interesting question to ask is the following.
How many distinct weak homotopy types do these nine topologies represent?
A topology on a finite set is equivalent to a preorder (the specialization order). Moreover, any preorder which has a greatest or least element is contractible (take a homotopy which abruptly shifts from being the identity to mapping everything to the greatest or least element). Also, any finite space is locally connected, with its connected components being the same as connected components of the associated preorder.
Now, it is easy to see that for any preorder on a 3-element set, each connected component has either a greatest element or a least element. Thus any topology on a 3-element set is homotopy equivalent to a discrete space, so the homotopy types are just those of discrete spaces with 1, 2, or 3 points (corresponding to the number of connected components of the preorder).
Some brief remarks on the general theory: the weak homotopy type of a finite space is that of the nerve of its specialization preorder, and every finite simplicial complex is homeomorphic to the nerve of a finite poset, so classifying finite spaces up to weak homotopy type is as hard as classifying finite simplicial complexes up homotopy type (in particular, it is not computable). Homotopy types of finite spaces correspond to finite posets which have no elements with a unique successor or predecessor (a unique successor being a least strictly greater element, and similarly for predecessors); every finite preorder can be uniquely reduced to one of these by first taking the antisymmetric quotient to get a poset (which does not change the homotopy type) and then repeatedly identifying elements with their unique successors or predecessors (which again does not change the homotopy type). It follows that the homotopy types of spaces with $n>0$ elements are in bijection with the isomorphism types of such nonempty posets with at most $n$ elements.