When the opponents open 1NT it is best to overcall only on shapely hands (especially over a strong NT). Since you are already at the two level, you really need something more sophisticated than simple natural overcalls in order to show two suited hands. There are numerous different defences to 1NT around, but one of the most popular by more advanced players in America is Cappelletti (also known as Hamilton – depends on which coast of the USA you are on).
This defence is not common in Europe; but a popular defence is Multi-Landy, which is the same as Cappelletti except that the meanings of 2 and 2 are reversed. I believe that Multi-Landy is a definite improvement over Cappelletti and so it is what I shall describe here.
Playing Multi-Landy, over a 1NT opening by opponents the bids are :-
Now let’s cover what the partner of the Multi-Landy bidder does. We’ll call them responder.
Over 2, responder bids their best major or 2 if they are equal length. This is the big advantage over Cappellitti - when responder is equal length they bid 2 and then the over-caller can bid their longer major. With Cappelletti it is a lottery as responder must choose a major and you can easily end up playing in a 4-3 fit when there is a 5-3 fit in the other major. Obviously when responder has one major longer than the other, they bid the longer.
2 is usually a major suit. If you have a decent minor suit, it will often pay to defend 1NT – especially if you are on lead with a suit that is good to lead from. Also, if your suit is a minor, then you have to play at the 3 level so it is usually best to defend. Thus this 2 is usually (but not always) a major suit. Responder bids using the same philosophy as with a multi 2 opening that includes a both majors option, for example:-
Responder normally bids 2, but bids 2 if they are prepared to go to 3 or more opposites a heart suit (the same principle as used in the Multi 2). Bear in mind that over-caller is limited to around 14 high card points (with more, they would usually double). Of course, nothing is lost on the odd occasion where over-caller has a long minor (you will always be at the 3 level).
Over a 2 of a major overcall, responder normally either passes or bids 2NT which over-caller will correct to their minor. If responder bids their own suit then that is simply to play (presumably a 6+ card suit). A raise to 3 of partner's major is invitational to game.
Double should be passed unless responder has a very weak hand (4- points), in which case they should attempt to bail out by bidding a 5 card suit. With a very flat weak hand it is usually best to pass; partner may just have 1NT set in his own hand, and even if 1NT doubled makes it is usually not total a disaster (it is not a game contract). Note that the double should have an upper limit of about 18-19 points. Now it would be nice to defend 1NT doubled when you have a huge hand but partner is very likely to have less than 5 points and will pull it!
Multi Landy has the (very big) advantage that you always find the best major suit fit. Cappelletti has the advantage that you can play in 2 (as opposed to 3) with a single suited hand if you prefer not to defend 1NT. Up to you.