If you have a rear rack and some bungee cords, you can haul your wash to the laundromat. If you're not used to carrying a load and are wary of the idea, try it out in your driveway and see how your bike handles - be sure to go more slowly and give yourself more space to turn.
First, get a sheet or towel to bundle your laundry in, then tie the ends of the bundles tight.
The rack on my bike has a spring arm which is nice but not essential. Position the largest bundle and crisscross bungees around to secure it on the bike and to hold the knots in place. Add additional bundle, repeat. Make sure everything is secured and off you go.