I do not have a definite answer but rather some reflections I've being doing myself and with a colleague I'll mention later, recently, on the subject.
Sure Poisson cohomology can help you in detecting the"missing" leaves, my favourite example being the triple of bivectors $\partial_x\wedge\partial_y$ (trivial 1-Poisson cohomology),
$(x^2+y^2)\partial_x\wedge\partial_y$ (1-dimensional 1-Poisson cohomology), $(x^2+y^2)^2\partial_x\wedge\partial_y$ (infinite-dimensional 1-Poisson cohomology). In a sense $1$-dimensional Poisson cohomology represents the tangent space to the set of leaves (which is a badly behaved non Hausdorff space and may therefore have no nonconstant functions defined on it) and is therefore slightly more sensible.

<p>Now, you may know that Poisson cohmology may be defined also with values in a Poisson module. If $T$ is a distribution (I mean a linear continuous functional on $\cal C_0^\infty(M)$ - compactly supported smooth functions) then letting $\left\{f,T\right\} (g)=T(\{f,g\})$ one gets that $\{f,T\}$ is still a distribution. One can show that in this way distributions form a Poisson module over $\cal C_0^\infty(M)$. One can of course look for the annihilator of such module, i.e. Casimir distributions defined by $\left\{f,T\right\}=0$ for every $f$. This space enlarges naturally the space of Casimir functions. On the quadratic singular Poisson structure I was referring to above it is possible to show that the vector space of Casimir distributions is $5$-dimensional generated by the $\delta$ function of the origin, its first derivatives and its mixed second derivative. </p>

One could consider this space of Casimir distributions, or more generally the Poisson cohomology with coefficients in this Poisson module. This I've never attempted to compute, even in easy examples.


I've basically learned this by Paolo Caressa who wrote a couple of notes about this:

Examples of Poisson Modules, I, Rendiconti del Circolo Matematico di Palermo(2) 52 (2003), 419-452

Examples of Poisson Modules, II, Rendiconti del Circolo Matematico di Palermo(2) 53 (2004), 23-60. 

The examples above are all $2$-dimensional. It is important to recall that every $2$-dimensional bivector is Poisson and determined by a smooth functions on the plane. We are therefore looking for something that should be, in this specific case, distinguish singularities of smooth functions, quite subtle, therefore.

Something easier, though not so capable of detailed analysis is to consider a maximal subalgebra of continous functions to which the Poisson bracket may be extended. This was done, for example, in an old paper by Albert Sheu on the quantization of the Podles sphere (the one with an appendix by Lu-Weinstein, I do not have a chance to get the exact reference now) where he quantizes the algebra of all smooth functions on the sphere minus the North Pole that can be continously extended to the North Pole. The reason why this is much less sensible is that as long as you have quadratic singularities in the bivector you can had square root singularities in the functions, which immediatly throws in all continous functions. 

Hope this may help.