If you want to be eligible for Mayorships, make sure you aren't checking in privately. Private check-ins will not count towards the Mayorship tally nor will they unlock a Mayorship. When you check in, make sure to select Yes to "Share with friends?" and you should be able to unlock a crown. Only users who have legit check-ins (no rapid-fire check-ins, no drive-by check-ins, and no fake check-ins) will be able to reign.
In order to be eligible, the setting "Let me earn mayorships (I realize that mayorship is a public office)" must be checked and you must have a profile photo. You can upload a photo to your account on the app or at http://foursquare.com/settings. Click "edit" next to Account Info. **Photos must be 200 KB or smaller (jpg, gif, or png format).
Remember that Mayorships are based on the most *days* with check-ins in the past 60 days (you could have more check-ins overall but fewer check-ins over the past 60 days). It's also possible that you're neck and neck with another user who is checking in to the venue often, in which case that's all part of the fun!
Oh, and if you’re using the mobile website at m.foursquare.com, your check-ins will not count towards the Mayor tally.