What does law enforcement look for when hiring females?

If you can pass both the physical test, and make it through the acadamy, you should be fine. I am pretty sure that the expectations are alighly different for a man than a woman, but not in an unfair way. Your status as a single mother should really have no bearing on your employment chances at all.

It shouldn't matter that you're a woman. I think as long as you are qualified for the position in which you are applying, and so long as you are physically capable to do the job..You shouldn't have any problems with it. They shouldn't discriminate against you because of your gender (or the fact that you have a special needs child - that shouldn't matter).