A lot is known about geometric and topological properties of diffeomorphism groups of surfaces (here, I am mainly thinking about the work of Smale and EellsElworthy). Is there anything known for orbisurfaces ? My first guess would be that many of these groups must have contractible components since singular points impose extra conditions somewhat similar to fixed points. Is there a good reference on this topic ?
The result you want can be found in the following paper:
MR0955816 (89h:30028) Earle, Clifford J.(1CRNL); McMullen, Curt(1MSRI) Quasiconformal isotopies. Holomorphic functions and moduli, Vol. I (Berkeley, CA, 1986), 143154, Math. Sci. Res. Inst. Publ., 10, Springer, New York, 1988.
What they prove is actually pretty remarkable. Namely, let $S$ be a hyperbolic surface. Then there is a family $\phi_t$ of selfmaps of $\text{Diff}^{0}(S)$ such that $\phi_0$ is the identity, such that $\phi_1$ is the constant map taking each diffeomorphism to the identity diffeomorphism, and such if $f \in \text{Diff}^0(S)$ commutes with a finite order diffeomorphism $g$ of $S$, then $\phi_t(f)$ also commutes with $g$ for all $t$. In other words, you can contract $\text{Diff}^0(S)$ in way that doesn't break any symmetries.
Now assume that $\Sigma = S / \Gamma$ is a good hyperbolic orbifold, where $\Gamma$ is a finite group of diffeomorphisms of $S$. The identity component of the orbifold diffeomorphism group of $\Sigma$ is then homeomorphic to $$\text{Diff}^{0}(S,\Gamma) := \langle f \in \text{Diff}^{0}(S)\ \ gfg^{1}=f\ \text{for all}\ g \in \Gamma \rangle \subset \text{Diff}^{0}(S)$$ The nullhomotopy $\phi_t$ preserves $\text{Diff}^{0}(S,\Gamma)$, so it is contractible.
(EDIT : I made a slight fix to the definition of the orbifold diffeomorphism above. It doesn't change the argument. Thanks to Tom Church for pointing it out to me!).

$\begingroup$ Thanks for this quick and illuminating answer. Do you know if some of those results extend to nonhyperbolic orbifolds ? $\endgroup$ – Martin Pinsonnault Jul 7 '10 at 19:51

$\begingroup$ They fail even in the classical case for nonhyperbolic surfaces (for instance, for the sphere or the torus). $\endgroup$ – Andy Putman Jul 7 '10 at 19:54

$\begingroup$ Well, my question was badly formulated. Here's a second try: Do you know any results on the homotopy type of diffeomorphism groups of nonhyperbolic orbisurfaces (à la Smale), regardless of the techniques used ? $\endgroup$ – Martin Pinsonnault Jul 7 '10 at 19:58

$\begingroup$ Sorry, I'm not aware of any such results. I'd be interesting in hearing about any ones that you find! $\endgroup$ – Andy Putman Jul 7 '10 at 20:11