When you upload a map to your fileshare is there a way to see how many times it has been downloaded by people?
There currently is no way to do so. It might be tracking downloads to display them at a later date, but nobody knows (outside of 343i that is).
yea I would really like that too... but keep in mind that if they are tracking it behind the scenes to display at a later date, uploading your file with new versions each time probably means that the counter resets. there is probably a unique ID assigned to each file that sits in a database somewhere, so each file has it's own count. for now, a good read on popularity of a map might be the number of views or responses on the main map thread, which you can sort by both views and replies.