Would I ever open 1NT with a singleton? Maybe once a year. Consider this hand:

 K  
KQ65  
 AQ2  
Q8762.
 

An opening of 1 is "normal," but what next? What if partner responds 1 (that wouldn't be a surprise, since you have only one spade). What would your rebid be? Let's look at the possibilities:

1) Rebid 1NT. This would show 12-14 balanced. You don't have 12-14 and you aren't balanced.
2) Rebid 2. Typically this would show 6+ clubs, but it is a possible rebid with this hand.
3) Rebid 2. This is a reverse and shows at least 16/17+. You aren't worth it.

So, rather than commit one of those misdescriptive rebids, it is possible to open 1NT to begin with. There is nothing illegal about it. You aren't opening it to be a deviant or trouble-maker. It is just a judgement call--and starting with 1NT might be the best lie.

What if you open 1NT with the hand above and partner (you know how they can be) transfers to spades (he bids 2)? You just have to suck it up and bid 2. Sure, if your partner passes, it will be a 5-1 fit, but it isn't the end of the world. Often, your partner will bid again and you can end up in a different contract. Worst case, tell your partner you had a club in with your spades.

Note: I wouldn't ever open 1NT with a small singleton. If in the hand above, the K were a low one (and the 3 points were elsewhere), I'd open 1 and rebid 2 (as the least-bad lie).

I don't go out of my way to open 1NT with a singleton. I would do so if and only if it was to avoid a potentially awkward rebid problem (as with the sample hand above). Switch the majors (

 KQ65  
 K  
 AQ2  
 Q8762)

and I would gladly open 1 (knowing I could bid spades next without having to reverse).

Note: Before August, 2016, some ACBL clubs disallowed 1NT openings with a singleton. Now, the rule states exactly: "A notrump opening or overcall is natural if, by agreement, it contains no void, at most one singleton which must be the A, K or Q and no more than two doubletons. If the hand contains a singleton, it may have no doubleton.”