A note before: the communities of low-dimensional topologists and categorically minded people have different opinions on what an
'orbifold' is. Let $G$ be a discrete group that acts properly and smoothly on the manifold $M$. As far as I understand,
low-dimensional topologists consider the quotient space $M/G$, together with the cleverly packaged information on the isotropy subgroups.
For a categorist (and I am following them on this issue), an orbifold is a special type of 'stack', which in turn is an object of a
suitable category of functors to groupoids (and can be represented by a certain Lie groupoid; in the case under discussion, it is the action
groupoid of the $G$-action on $M$).
I think of an orbifold as a space whose points can have automorphisms and (as a side remark) my personal opinion is that it is quite unfortunate that
there does not seem to
be a definition of a stack that does not appeal to the functor-of-points-philosophy.

Now to the real issue I wish to communicate.
The Pontrjagin-Thom construction is fundamental in most branches of manifold topology. The basic case is when $f:E \to B$ is a proper
smooth map of manifolds, $dim (E)=dim(B)+d$, $d \in \mathbb{Z}$. There is a stable normal bundle $\nu(f):= f^* TB - TE$ and the
PT-construction is a map of spectra $\Sigma^{\infty} B_+ \to Th (\nu(f))$, the Thom spectrum of $\nu(f)$. From this and the Thom isomorphism, one deduces homological invariants, such as the Gysin map $f_!: H^i (E) \to H^{i-d} (B)$, which is moreover functorial , $(f \circ g)_! = f_! \circ g_!$.

My initial thought was that there should be an analogue for orbifolds. The tangent bundles make sense, and one can make sense
out of stable vector bundles on a stack. Moreover, there is a homotopy type of stacks. If the stack is $M//G$, its homotopy type $Ho(M//G)$
is the Borel construction $EG \times_G M$, and it comes with a map to $M//G$ that is declared to be a weak homotopy equivalence. One can then define homotopy/(co)homology of stacks using this
homotopy type (there are finer theories, due to Henriques-Gepner).

One can pull back stable vector bundles on stacks to the homotopy type and form the Thom spectra there. The expectation is that there
is a PT map $\Sigma^{\infty} Ho(B)_+ \to Th (\nu(f))$ when $f$ is proper in a suitable sense. The unique map
$\ast //G \to \ast$ should be a proper map.

Unfortunately, this does not work. Together with Jeffrey Giansiracusa, I wrote a paper that gives such a PT-construction, under the
restriction that the map $f$ is ''representable'', which in the case $B=\ast$ amounts to saying that $E$ is a closed manifold.
In his study of orbifold cobordism, Andres Angel used a different approach than the PT-construction.

Why is such a construction impossible? Well, let us suppose that there is such a construction that works at least for $\ast//G \to \ast$
(the quotient map $\ast \to \ast //G$ does not pose a problem). As in the case of the ordinary PT-construction, one defines the Gysin maps $f_!$ and proves that they are functorial. As the composition $\ast \to \ast//G \to \ast$
is the identity, the composition of the corresponding Gysin maps must be the identity. On the other hand, we know very well what the Gysin
map of $\ast \to \ast//G$ does on $H^0$; it is multiplication by $|G|$. This leads us to the contradiction that the identity on $\mathbb{Z}$ factors
through the multiplication by $|G|$ map.

Strangely enough, when we take real coefficients, the Gysin map is integration over the fibres, and this seems to work for orbifolds.
I can also conceive an integration map in complex $K$-theory, but nevertheless the problem seems to be fundamental and severely restricts the application of manifold methods to the topology of orbifolds.