The paper considers pairs $(X,B)$ where $X$ is a normal projective surface over $\Bbb C$, and $B$ is a $\Bbb Q$-divisor whose coefficients are $1$ or $1-1/m$ for some natural number $m$. A log canonical singularity on such a pair is a quotient by a finite or infinite group, so if $(X,B)$ has log canonical singularities, the orbifold Euler number $e_{\rm orb}(X,B)$ can be defined. The main result is a Bogomolov-Miyaoka-Yau-type inequality which implies that if $(X,B)$ has log canonical singularities and $\kappa(X,K_X+B)\ge 0$ then $(K_X+B)^2\le 3e_{\rm orb}(X,B)$. The actual inequality proved is somewhat stronger and it also implies all the previously published versions of the Bogomolov-Miyaoka-Yau inequality. The proof involves the Log Minimal Model Program, $Q$-sheaves when $K_X+B$ is nef, and a study of the changes in the two sides of the inequality under a contraction. The paper also contains a further generalisation where the coefficients of $B$ can be arbitrary rational numbers in $[0,1]$, a different condition is imposed on the singularities and $K_X+B$ is required to be nef. Some applications of the inequalities are also given, for example, estimating the number of singularities or certain kinds of configurations of curves on surfaces.