You have two questions there. As to what to wear around the ship - almost anything goes. Believe me, I have seen people wearing things that most people would not wear outside their own house. However, as long as all the right parts are covered up, nobody is going to say anything.
As to the dining room on non-formal nights, you cannot wear tank tops, shorts or bathings suits, and you cannot be in bare feet.
The following is from Royal Caribbean's website ...
Q: What are the dress codes onboard?
A: There are three distinct types of evenings onboard: casual, smart casual and formal. Suggested guidelines for these nights are:
Casual: Sport shirts and slacks for men, sundresses or pants for women
Smart Casual: Jackets and ties for men, dresses or pantsuits for women
Formal: Suits and ties or tuxedos for men, cocktail dresses for women
Onboard attire...
Shorts, t-shirts, slacks, trousers, casual skirts, blouses and sundresses are all perfect during the day, whether indoors or out.
For lounging by the pool, pack two bathing suits (an extra suit in case the other is wet) and a pair of casual shoes (e.g., flip-flops, sandals, sneakers). You'll also need some casual shorts and a couple of t-shirts or a cover-up for getting to and from the pool areas.
If you're planning to visit our onboard fitness center, be sure to bring extra shorts, t-shirts and socks that you won't mind working out in - plus a pair of tennis or running shoes/sneakers.