Sorry saublewwater - not even travel irons. But if you want to use your curling iron you can. Probably the straightening iron would work better though.
Like Sonny said portable steamers work well.
Here's the policy:
Guest Conduct Policy Prohibited Items Items with Heating Elements or Open Flames Certain items that generate heat or produce an open flame are not permitted on ship. This includes clothing irons, hotplates, candles, incense and any other item that may create a fire hazard. (Curling irons and hair dryers are allowed.)