If $X$ is a compact Kahler manifold, then Hodge theory says that its cohomology decomposes as a direct sum
$$ H^{p+q}(X,\mathbb C) = \bigoplus_{p,q} H^{p,q}(X,\mathbb C) $$
where $H^{p,q}(X,\mathbb C) = H^q(X,\Omega_X^p)$ are the Dolbeault cohomology groups and $H^{p,q}$ and $H^{q,p}$ are conjugate isomorphic. One can prove that the same holds for a Fujiki manifold, i.e. a complex manifold bimeromorphic to a Kahler manifold.
Q: What happens for compact complex spaces?
Here "complex space" should mean singular and non-reduced. Is there a notion of a "Kahler" complex space, on which some predictions of Hodge theory hold? The question should make sense, as we can define $H^{p+q}(X,\mathbb C)$ topologically and the $H^q(X,\Omega_X^p)$ exist for a complex space $X$, but do we even know that the latter are subgroups of the former for some special spaces $X$?
One can imagine naively defining such spaces as those which admit a Fujiki desingularisation (Kahler to begin with, but desingularisations are not unique so we need to compare them somehow), and then hoping that the "upstairs" decomposition induces one "downstairs". Of course, if it were that simple someone would have done it ages ago.
Best Answer
Let me add to Donu's mentioning Du Bois's Hodge decomposition. First of all, many feel that part of the credit is due to Deligne as Du Bois built heavily on his ideas. Then again that is probably true for many things in Hodge theory.
Anyway, Du Bois's main idea was that one can do Deligne's construction "one step earlier" in the sense that Deligne used simplicial resolutions to build his Hodge structure on the cohomology groups, and Du Bois does this in the derived category of coherent sheaves (with some conditions...) so he obtains a filtered complex (now mostly called the Deligne-Du Bois complex) which is quasi-isomorphic to the constant sheaf $\mathbb C$ and whose associated graded quotients (these are actual quotients in the category of complexes, before one passes to the derived category) are the objects $\underline{\Omega}_X^{p}$ Donu mentions.
It turns out that it follows directly from the construction that there is a Hodge-to-de-Rham (a.k.a. Frölicher) spectral sequence and from Deligne's work that it degenerates at the $E_1$ level. So, a lot of things actually work out the same way as in the smooth case if one uses $\underline{\Omega}_X^{p}$ in place of ${\Omega}_X^{p}$. For instance, the Kodaira-Akizuki-Nakano vanishing theorem also holds (as well as the existence of the Gysin map, etc) that can be used to prove a singular version of the Lefschetz hyperplane theorem cf. here.
As Donu mentioned, it is not known in general when the natural map ${\Omega}_X^{p}\to\underline{\Omega}_X^{p}$ is an isomorphism, except for $p=0$ which is the definition of Du Bois singularities. I think in general (that is, for arbitrary $p$) it is true that for toroidal singularities this is an isomorphism, or at least there is a good description of each object and the map between them. We have a pretty good understanding of Du Bois singularities, although not complete and there are still many interesting open questions. Rational singularities are Du Bois by this, log canonical singularities are Du Bois by this. For an intro to Du Bois singularities and related stuff you can try this.
There is also an intriguing connection to singularities defined via the action of Frobenius in positive characteristic. For more on this see this and other works of Karl Schwede.
Regarding the question on having something that reflects the non-reduced structure, I am afraid that the topological $H^i$ do not see the non-reduced structure, so I don't think you can expect any reasonable Hodge theory that remembers that.