Competitive bidding

1NT(X)

I suggest Terrorist’s runout regardless of what (X) means. This runout also applies to a conventional double, yet responder can also pass balanced hands more often.

Opener often runs with clubs because it is hard for responder to do so.

1NT(2♣)

Systems on is a decent approach over 1NT(2♣︎). The overcall is somewhat ignored, and X replaces the stolen bid of 2♣. Since continuations to 1NT is major suit oriented, it is crucial if (2♣) shows a major suit.

(2♣) shows both majors

This kind of 2♣︎ requires extra care. I recommend BTU vs Unusual.

(2♣) specifies one major

This position is analogous to (1M)1NT. X replaces the Gladiator 2♣.

(2♣) specifies no major

Ignore the (2♣) overcall. X replaces the Stayman 2♣ as usual.

1NT(2♦♥♠)

Rubinsohl covers these auctions. Similar to Rubens advances, the transfer structure helps saving bidding space. Showing the suit faster also helps competition.

Swap 3♥♠ against (2♠) like Larry Cohen to avoid declaring hearts.

1NT(2NT)

BTU vs Unusual suffices.