However, lots of spaces are images and these generally have singularities. For example, the cross cap, seen on the What is a singularity? page has singularities. The cross cap is stable in the sense that small perturbations of the map do not remove the singular point. Furthermore, the singularities are non-isolated. Therefore this singularity is common (because it's stable) but not studied much (because it's not an isolated singularity)! What this means is that there are many, many problems that have not been looked at.
My work in this area has focussed on the use of a spectral sequence that computes the homology of the image of a finite map. There are plenty of unsolved problems just begging to be attacked by this tool.
We can play this game for lots of other maps. I have worked on those that occur unavoidably in one-parameter families of map. An obvious extension would be to two-parameter families. When this is done, one could produce Vassiliev type invariants.
However, a more fruitful investigation could be made of functions on singular spaces. For example, take a cross-cap or its higher dimensional generalization. What is the classification of functions on this set? A simple question yet little has been done in this area, despite the fact that one could get interesting geometrical results from it.
Definition of symmetry set of a curve: Consider a non-singular plane curve C. The symmetry set of C is the closure of the set of points of centres of circles that are tangent to C at two or more points. This measure the shape of the curve.
One can generalise to manifolds in any Euclidean space - just take the closure of the locus of points of spheres that are tangent at two or more points.
Interestingly, this gives old fashioned geometry - doing what the ancients wanted to do but didn't have the tools. Too much geometry these days doesn't involve striking visuals - or even any pictures. Researchers give talks where they fill the blackboard with &nabla after &nabla. These are results that the ancients such as Monge, Cayley and Gauss would say 'How cool is that?' or whatever phrase academics used in those days to show their appreciation.
Another reason for it being interesting is that it is close to applications in the real world. Anything you do here has the potential to be used in real-life pretty soon. (My usual stuff on the topology of high dimensional complex analytic sets is unlikely to be used soon - if ever!)
Lots of different areas of research are possible. Simple questions do not seem to be answered. What is the symmetry set of the symmetry set is an amusing one -- it does assume that we can define the symmetry set of a singular set, but that, again, is an unexplored topic. Much work is in its infancy, so for example, only recently has Jim Damon given a reasonable definition of the Weingarten map for medial axes.
So, plenty to do!