See a theorem of Richards, which implies that homeomorphism types of planar surfaces are in 1-1 correspondence with homeo. types of compact subsets of the Cantor set. I think there should be uncountably many homeo. types of totally disconnected compactums, but I don't know a reference or an argument off the top of my head. I think this should be related to the ordinalities of the accumulation points, but I'm not sure which ordinals can occur.

Addendum: Googling, I found references to a result of Markiewicz-Sierpinski classifying
countable compact metric spaces up to homeomorphism by their Cantor-Bendixson rank (see section 3 of this paper for a statement).
The CB-rank must be a countable ordinal $\zeta$, and the space is homeomorphic to the
ordinal $\omega^\zeta\cdot n+1$ with the order topology for some $n\in \mathbb{N}$. These may all be realized as
compact subsets of the line. This gives uncountably many non-homeomorphic compacta,
which by Richards' theorem implies that there are uncountably many planar surfaces.