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It really depends on the ship & cruise line. Almost every ship has an option to visit perhaps a "buffet-style" restaurant where you will generally be OK with casual clothing. In the main restaurants there will usually be a different dress-code each evening (e.g. Casual, Informal & Formal). Casual will normally be trousers & shirt for men (no jeans/trainers) and trousers or skirt & blouse for the ladies. Informal will normally be Jacket & Tie (or business type suit) for men and cocktail dress or trouser-suit for ladies. Formal will mean dinner suit/tuxedo for men and formal dress for the ladies.
If you do not wish to be bound to these rules, Ocean Village advertise to be for "people who don't do cruises' which can be taken to mean that dress-codes are not enforced.
I would like to say from a personal point of view though that I am 100% a Jeans & T-Shirt person at home and therefore when cruising, I love having to dress-up in a way I would not normally do at home.
Hope This Helps.
The cruise line we use a lot is Celebrity and they have usually on a 12 day cruise 3 formal / 3 informal which is jacket and tie and the remainder are casual. Even on the formal nights many now just have a lounge suit, I do posses a tuxedo but on our last cruise out of around 1800 passemgers only about 40 had one and a large number had dinner suites but I guess around 25% wore a suit or a nice casual oufit. Most female passengers do like to dress up whatever the dress code is for the evening. Most ships apart from the main dining room have more casual restaurants so never a need to be concerned, so long as you enjoy your trip.