Let $X$ be a smooth surface of genus $g$ and $X^{[n]}$ its n-symmetrical product (that is, the quotient of $X \times ... \times X$ by the symmetric group $S_n$).  There is a well known, cool formula computing the Euler characteristic of all these n-symmetrical products:

$$\sum_{d \geq 0} \ \chi \left(X^{[d]} \right)q^d \ \ = \ \ (1-q)^{- \chi(X)}$$ 

What about for singular surfaces?  More precisely, if $X$ is a singular complex algebraic curve, do you know how to compute the Euler characteristic of its n-symmetrical powers? (or yet more to the point: the Euler characteristic of the space of 0-subschemes of length n of $X$)