These notes contain the definition of various chain complexes and groups (co-homology/homotopy) that are in-principle computable. However, I have not seen this being efficiently applied somewhere, and not been able to come up with interesting applications myself. Topics I have considered are:
1. Triangulation of solutions of equations (for e.g. homology computations) does not seem to be very popular, at least in dimension >3. I suspect this is a hard problem, but maybe I am just missing pointers to the right literature.
2. CAD applications or Computer Games have lot's of triangulated objects. Their topology seems to be not to be very interesting. Again: In dimension <=3 there is really not that much going on. And since you have constructed the object yourself, you probably know the geometry already.
3. Graphs appear everywhere, and can be viewed as 1-dimensional chain complexes but do not have interesting co-homology groups.
4. Did anyone compute homology groups for "Manifold Learning"?
It's also not clear to me, how much interesting information can be extracted from those homology groups. Applications of Homotopy/Homology in (semi-classical) Physics are already quite slim (apart from Quantum Field Theory, String Theory, Gauge Theory?!) as most of it takes place in contractible spaces $IR^n$.