Why would Polonius ask Reynaldo to find out what the town thinks of his son in Hamlet?
There are two ways to interpret this very interesting scene. Firstly, it is possible to see this as Polonius being, yet again, over-officious and not leaving anything to chance. He is shown to be such the servant to Claudius, and his strengths lie in his punctilious nature. Therefore he wants to leave nothing to chance and wants to make sure his son is doing nothing to bring harm upon himself, or at the very least he wants to ensure that if Laertes is indulging in such behaviour, Polonius knows about it first. This is of course infered, because even when Reynaldo asks Polonius why he is spying on Laertes, Polonius responds in a way that paints him as a figure of fun who can't remember the reason:
...what was I about to say? By the mass, I was about to
Say something. Where did I leave?
This view supports the idea that Polonius is a bumbling father who is losing his faculties, but who means well, and perhaps wants to keep an eye on his son as part of expressing his good intentions towards him.
The second view is to see Polonius as a manipulator and politic intriguer. The fact that he sends a spy on his son says a lot about his character and how he automatically distrusts even those closest to him. It also prepares us for his role in going as far as using his daughter as bait in his plot to expose the real reasons for Hamlet's madness. Note the language he uses to Reynaldo on what his real job is with Laertes:
And thus do we of wisdom and of reach
With windlasses and with assays of bias
By indirections find directions out.
Such language does not come from a man who is faltering and doddery, but from a calculating and dangerous mind who himself is not averse to presenting himself as one innocuous character only to obscure his much more dangerous self.