I will be writing up your answer as a short paper. Please get in touch if you would like to be listed as author, otherwise I will cite your answer here. You can reach me as janus@<domain listed in my profile>.

Thank you very much, BS! Perhaps if I had not been so convinced that all curves of critical points would extend through the any singular points, I would have had better luck finding a counterexample myself :) Thus I learn again that (my) intuition and analysis should not be mixed. This is really amazing -- I have been pushing this problem to mathematician friends for a couple of years now with no progress, and then a few days after posting it to mathoverflow it is solved. The future is now (but then, it's 2010 -- so it'd better be) :)

@BS Thank you very much -- this is getting very close to what I was looking for. Could you recommend a good reference for somebody with only rusty undergrad analysis and differential geometry as background? Do I understand your last statement correctly that your argument does not guarantee that the curves extend smoothly across the singular points, or is it clear that there are situations in which this the curves would not extend smoothly across the singular points?

@Will Jagy: Yes, it is certainly possible to construct intersections of analytic curves in this way, although some constraints are imposed by the harmonicity requirements (see arxiv.org/abs/0802.3162 for the case of two curves). What my post pertains to is in some sense whether this approach is complete: i.e. whether all possible intersections of guide curves are intersections of analytic guide curves. You might have to spell out a bit the implications of $\vec{C}\cdot\nabla$ commuting with $\nabla^2$ :)

@Willie Wong: Thank you very much for pointing this one out. I have corrected my definition of guide curves so it now hopefully describes what I am looking for.

Background: The question is relevant to networks of rf ion traps, where the trapping potential is the ponderomotive potential associated with an oscillating electric field. The local amplitude of electric potential oscillations is described by a harmonic function, and for practical reasons it is preferable to trap ions on critical points of this function so that transport would have to take place along guide curves as introduced above. The aim of my question is to establish what intersection topologies are possible for guide curves.