With any tramp, you need to be careful not to use too harsh a chemical cleaner. There are many household products that will clean the tramp, but damage the fabric (coating), and over time, develop holes and even deteriorate to the threads (guess how I know)...
My cleaner of choice is Dawn, if for no other reason than it's biodegradable and safe enough that it's used to clean birds after oil spills. It cleans grease and road grime like nobody's business. Not sure how effective it is on mold, but I would start with Dawn before moving on to anything harsher.
Hope this helps.