Intersection homology is alive and well in a large number of guises. It's true that a lot of the work trended to algebraic geometry, representation theory, and categorical constructions, such as perverse sheaves, through the 90s, but there also continues to be work in the more topological settings by people such as me, Cappell, Shaneson, Markus Banagl, Laurentiu Maxim, and many others. At least some of this work is dedicated to extending classical manifold invariants, such as characteristic classes, in a meaningful way to stratified spaces, such as algebraic varieties, and there is a lot of recent interest (though slow progress) in figuring out how intersection homology might tie into various algebraic topology constructions. There are also analytic formulations such as L^2 cohomology (initiated by Cheeger), and much more.
Here are some good references to get started in the area:
An Introduction to Intersection Homology by Kirwan and Woolf (mostly concerned with telling the reader about the fancy early applications to algebraic geometry and representation theory, but a great overview nonetheless)
Intersection Cohomology by Borel, et.al. This is a great serious technical introduction to the area and, to my mind, the canonical source for the foundations of the subject)
Topological Invariants of Stratified Spaces by Markus Banagl (topological but mostly from the sheaf point of view)
For an overview of state-of-the-art in intersection homology and related fields, I'm co-editing a volume on Topology of Stratified Spaces that will be published in the MSRI series. Unfortunately, it's not out yet, but look for it soon.
The original papers of Goresky and MacPherson are quite good.
Topological invariance of intersection homology without sheaves by Henry King is a good introduction to the singular version of the theory.
And for a whole pile of recent papers, I'll shamelessly plug my own web site: http://faculty.tcu.edu/gfriedman/
and Markus Banagl's: http://www.mathi.uni-heidelberg.de/~banagl/
And many further references can be found from these locations.