Take-Out Double over 1NT

Let's take the most frequent example we meet in the UK. Opponent opens 1NT 12-14 and partner has doubled showing middle of range or more - 13+. This 13 HCP critical value may be adjusted according to the openers strength. E.g. a 3rd/4th seat 1NT opening which is shaded to 11, double is shaded to 12! With a 14-16 opening use 15 etc.

Let's first remember what partner who doubles does NOT have

With a 6+ card suit & 13+ HCP as then Rubinsohl transfers are used - bid 2N, 3C, 3D, 3H to show C, D, H, S usually with good guards & a strong 6+ card suit). Also 3S-transfer to 3NT to play!
Two 4+ card suits as he would have bid the lower ranking - CDH <=13(opps max) HCP Bottom limit depends on vulnerability and lengths of the suits concerned. In any distributional hand we also prefer CDH with 13-14 HCP too
Nor will he have a 5 card Spade suit - he then bids 2
Also in the case of 5-4 with 5 Spades partner will still prefer 2 of the lower ranking suit CDH with 13HCP too

While one knows partner has 13+ HCP, one can also usually expect a somewhat flattish distributions

Responder's options following a T-O double:

Partner is certain to be balanced.

PASS. with 8-11(12) and no 5+ card suit seriously consider converting partner's double into a penalty - extrememy valuable when opponents are vulnerable!
With 11-12 and a good 5 card major suit bid 3 & game with a King value more!
Bid 2NT/3NT with (12)+ - let's take control or game with more! You will know where all the adverse points are!
Otherwise bid your best suit when not able to stand the double

Much Judgement is clearly required in this position:
The question that needs answering is this:-
Partner's double has thrown down the gauntlet and your dilemma is:

Safety first: escape the double hope for not worse than -50 or -100 or even a +90 or +110 win
Exert a penalty: 100 NV or the magical 200 V
Take control:: expectancy of 140+ when NVul.

Any subsequent double by either defender following a sequence begun ...1NT-*....is a mandatory penalty double.