No, cruise ships do not provide curling irons, however they are allowed onboard on most major cruise lines.

Most of the major cruise lines like Royal Caribbean, Celebrity, Disney, Cunard, Carnival and Viking do allow you to bring your curling iron or flat iron. 

There may be a requirement that the appliance (hair dryer, hair straightener etc.) have an auto shut-off feature to prevent an accidental fire.

Make sure that you are not using your curling iron near the heat detector in your cabin. It may seem far fetched, but there have been reports of cruisers who have done so and it has earned them a visit from security.

If, for whatever reason, the cruise line determines that your device might pose a hazard, it will be removed from your luggage and returned to you prior to your disembarkation. 

Please check with your specific cruise line for exact requirements.