  There exist several homology theories for singular spaces that satisfy
generalized Poincar\'e duality, including Goresky-MacPherson's intersection
homology, Cheeger's $L^2$ cohomology and the homology of intersection spaces.
The intersection homology and $L^2$ cohomology of toric varieties is known.
Here, we compute the rational homology of intersection spaces of complex
3-dimensional toric varieties and compare it to intersection homology. To
achieve this, we analyze cell structures and topological stratifications of
these varieties and determine compatible structures on their singularity links.
In particular, we compute the homology of links in 3-dimensional toric
varieties. We find it convenient to use the concept of a rational homology
stratification. It turns out that the intersection space homology of a toric
variety, contrary to its intersection homology, is not combinatorially
invariant and thus retains more refined information on the defining fan.
