There are lots of conventions out there for bidding over the opponents’ 1NT: DONT, Meckwell, Landy, Cappelletti, etc. Having an understanding of at least one of these conventions is very useful. But just as important as understanding how these conventions work is knowing when to use them.
There is a big difference between bidding over a strong notrump and a weak notrump. I like to define a strong notrump as any range that includes 16. So 14-16, 15-17, etc. are strong, whereas 12-14, 13-15, etc. are weak.
After a strong notrump, our goals are primarily to get in the way and compete for the partscore. We want to come in as often as we can to disrupt things. However, entering a notrump auction is inherently dangerous, which is why these conventions exist to let us show two-suited hands, which can be safer to bid with due to their higher ODR.
Vulnerability plays a huge role. Think of these bids a lot like preempts. You need to be very solid vulnerable, especially at unfavorable vulnerability; you can be friskier non-vulnerable, especially at favorable vulnerability.
Vulnerable you should be at least 5-5 in your suit; non-vulnerable you can be 5-4. Suit quality is more important than overall strength, though when you’re at unfavorable vulnerability you want a good hand to come in.
♠ QJ984 ♥ JT54 ♦ 654 ♣ 3
I wouldn’t dream of bidding with this hand when vulnerable. But I definitely would non-vulnerable.
♠ KJT65 ♥ AQ984 ♦ 9 ♣ 65
A solid overcall at any vulnerability.
Bidding is quite different over a weak notrump. Opener is only showing a minimum opening hand, so chances our side has a game increase considerably. It’s still most likely a partscore hand, but our priorities are not entirely obstructive; we want to compete effectively for the partscore, as it could well be our hand.
You want a full opening bid to come in over a weak notrump. Think of it like a 2-level overcall. You can use artificial systems to show some two-suiters. But you want to make sure your system includes a penalty double. A penalty double of a weak notrump shows a strong notrump hand or better. If you have a good suit to lead against 1NT, you can be a little lighter.
♠ KQT4 ♥ AQ4 ♦ QJ87 ♣ A4
An easy double.
♠ 65 ♥ AJ4 ♦ KQJT84 ♣ K3
With a good suit to lead, you can double on this hand with only 14 HCP.
If responder passes the penalty double, usually advancer will simply pass and go for a penalty. With a very weak or very shapely hand, advancer can pull the double. Bids are natural. Jumps show a good hand.
If responder bids over the double, we are in a forcing auction through 2♠. This means we cannot let them play at 2♠ or below undoubled; either we have to double them or outbid them. So a pass is forcing. The standard way to play is that after the penalty double, if they run our side’s next double is takeout; all subsequent doubles are penalty. So if they bid 2♥, natural, and you hold:
♠ Q4 ♥ AQT5 ♦ 98743 ♣ K3
You pass, and partner will probably double – a takeout double in the balancing seat – and you can convert that to penalty.