First, the city likely makes less than a sawbuck from every moving citation, so they are not doing it for the money. And if even 5% of those cited challenges the cite in court, then they will have to issue 20 more cites to break even. If they wanted money, they'd have all the officers writing parking tickets not movers.
Second, if you think you have grounds for a winning suit, articulate your damages and consult legal counsel. If an attorney is willing to take it on a contingency basis, you may be on solid ground. If he wants money up front or tells you that you have no suit worth pursuing, then you know your case is not all that good.
If all you want is to embarrass the city, you can always talk to the media or the city council and present them with your findings and hope that the law is on your side so that you are not made to appear the fool.

