Before we discuss conventions, let's discuss the situation.

To me, it is a crime to watch the opponents open 1NT and bid unimpeded. Your opponents are great bidders after their 1NT opening. They are so expert that I'll bet they use asking bids and relays. Yes, even your lesser opponents play that 2♣ asks and that 2♦ and 2♥ relay to the next suit. The science of bidding after a 1NT opening is quite accurate and all of your opponents are comfortable with the methods.

They are not comfortable, however, with interference. Not only do you take away their Stayman and Jacoby, but you put them into uncomfortable territory. Even experts with well-oiled mechanisms (lebensohl, negative doubles) have a difficult time coping with interference. If your partner opened 1NT (15-17) and you held, say, ♠54 ♥K9832 ♦K76 ♣K92, your plan would be easy. You'd transfer to 2♥ and then invite with 2NT.

But, if RHO overcalled 1NT with 2♠, would you be happy? What if the overcall was 2♦, showing diamonds and a major? What would your bids mean? Can you play from the right side? What would you bid? 

It's quite obvious how the interference makes notrump auctions a challenge. Given that, we should go out of our way (way out!) to interfere. Don't worry about "having your values," -- your goal is just to disturb, not to convey your hand to partner so that he can bid a game or double them.

Everything written here (whether against strong or weak) is played the same way in balancing seat as in direct seat (life's tough enough as it is).  In balancing seat (after 1NT P P ...), I can't stand to pass. I go out of my way to balance with almost any hand that contains a 6+ card suit, or at least 5-4 in any 2 suits. HCP don't matter (what you don't have, your partner will). I don't like to defend 1NT (with partner usually leading the suit I don't want led).

LC Standard uses DONT against a Strong Notrump, but different methods against a weak notrump (discussed further down).

A "Strong" Notrump is considered 14-16 or 15-17 or more.  If the bottom number of the range includes 13 (or 12,11,10, etc.), then it is considered a weak notrump.

Versus "Strong" (does not include 13 HCP)

Once you decide to interfere, I feel that the best method is DONT. There are many methods to choose from (so many, that you can start an alphabet song such as: A if for ASTRO, B is for BROZEL, C is for CAPPELLETTI, D is for DONT...)

DONT allows you to show all one- and two-suiters at the two level. Your goal should be safety. The other methods put more emphasis on penalty doubles, and or getting to majors. Furthermore, DONT is easy to memorize (very natural).

DONT

Opponents open strong 1NT:

In direct or balancing seat our bids mean:

• Double = One-suited hand (requests partner to bid 2♣)
• 2♣ = 2 suits, ♣ + higher
• 2♦ = 2 suits, ♦ + higher
• 2♥ = 2 suits, ♥ + higher
• 2♠ = Natural

(Against weak notrumps, I recommend some other system that employs double to show a good hand.)

Penalty doubles of strong notrumps are, in my opinion, horrible. Yes, horrible. It makes it easy for them to run out (often to two-of-a-minor, normally unreachable), and when they do sit, it's usually "impossible" to make an opening lead and defend (declarer's advantage).

Getting to the higher-scoring major is also not my priority; again, it's a matter of percentages. In the long run it's better to get all one- and two-suiters into play and not worry about how big your plus score is; the focus is on how low your minus score is! Get in, get out, and you'll be a winner.

Notes:

• Usually we try to have at least 5-4 in the 2 suits -- vulnerability is relevant, as is "position" (Direct seat is "never" 4-4).
• When vulnerable, you must have decent suits (not Qxxx and Jxxxx!). 
• Don't worry about game!
• Balance with "few HCP" but with shape!

Followups

After our Double: 

Partner "always" removes to 2♣. If the doubler now converts to 2♠ he is showing more than a simple 2♠ overcall.

After our 2♣ overcall: 

Partner should pass with 3+ clubs. With a doubleton club, partner will usually bid 2♦ to play in overcaller's other suit. 

After our 2♦ overcall:

Partner should pass with 3+ diamonds, unless he has both majors.  Otherwise, he should bid 2♥ to play in overcaller's major.  

After our 2♥ overcall: 

With equal length in the majors, partner can pass or hog the hand if he wishes. The most sensible idea is to pick the suit/side so that the strong hand (1NT opener) will be on lead.

What if we make a DONT overcall and their responder doubles?  For example: 

In this case, the partner of the DONT bidder does as follows:

• PASS = Willing to play in that suit (in the example auction, clubs).
• REDOUBLE = Not willing to play in that suit (asks partner to remove to his other suit)
• NEW SUIT = Natural--not asking for pass/correct

I advise only advanced players to bother memorizing the following (and even they probably should consider this not worth the memory strain as it "never" comes up:)

After all of these overcalls, 2NT (by "advancer") shows game interest and asks for more information. A 2♣ overcaller then bids 3♣ with a minimum, his other suit with a maximum. A 2♦ overcaller bids 3♣/3♦ with a minimum (tying to his major) and 3♥/3♠ with a maximum. A 2♥ overcaller bids 3♣/3♦ with a minimum (tying to his best major), 3♥/3♠ with a maximum.

All actions discussed above are alertable.

Versus Weak Notrump (contains 13 HCP)

For my intermediate (and lower) readers, I suggest you don't spend too much time on this. Very few of your opponents will be using a weak notrump. You'll be studying/memorizing something that "never" comes up. This defense to a weak notrump (especially dealing with doubles) does involve work and memorization. If you fear mixing it up with your Strong Notrump defense, then why even go here?

Against a weak notrump, LC STANDARD uses double as "penalty."  This means roughly 13+ -- any shape (if vulnerable, it will be a decent 13+). After the double, the partner of the doubler plays "Systems on" -- as if we opened 1NT.  For example:

2♣ is Stayman as if the auction went 1NT P 2♣ by our side.

X is negative as if the auction went 1NT (2♦) X by our side.

Just think of the initial double as meaning "I opened 1NT to show 13+" -- and proceed accordingly.  Advancer (the player opposite the penalty double) will typically leave the double in if he has a decent hand.  Once it goes (1NT)  Dbl (P)  P, they have been doubled for penalty, so all subsequent doubles are for penalty.

The other bids (as marked) against their weak notrump are "constructive." We don't interfere on garbage. On the other hand, these bids (below) are usually not made with a very strong hand (no double). So, these overcalls are roughly 11-16 in playing strength.

• 2♣ = Both majors (at least 5-4)  
• 2♦/2♥/2♠ = Natural (usually 6 cards)
• 2NT = minors

After the 2♣ overcall, 2♦=equal length in the majors, 2N=asking/game interest

This is a simple, easy-to-remember method versus weak notrump.  For players who are competing at a high level, facing a lot of weak notrumps, I suggest something a bit fancier.

Note: I am fully aware that this is only an outline. Professional partnerships need to do much more work. My suggestion, however, for 99.9% of players would be to just "wing it." In other words, agree on what's written here and just hope for the best at the table.