Sounds a little oxymoronic, doesn't it? A splinter normally shows a big fit for partner, a game-forcing hand, and a singleton or void in the bid suit. The splinter over 1NT certainly has the last two qualities, but we don't know about a fit yet.
We use 3H or 3S over partner's 1NT opening to show shortness in that suit and both minors. Some use this bid as 4441-type hands, but we prefer to have them show 5431-type hands. We'll open pretty much any in-range balanced hand with a five-card major with 1NT -- any 5332 and some 5422s. We've found that it's often right to play 3NT on these hands even with a 5-3 fit. The three times it's right to play a 5-3 major suit fit are
(a) You can ruff something from the three-card holding.
(b) You have an unstopped side suit that you can ruff to stop the bad guys from running tricks against you.
(c) You have a long (but not great) side suit in dummy that you need to ruff to set up.
Parts (b) and (c) don't usually come up when you've opened 1NT, so we worry about the first one. When responder has three cards in one major and a stiff or void in the other (and a game-forcing hand), we bid three of the short suit. Opener now knows (99% of the time) the best strain (if not the best level) for the partnership to play!
1NT - 3H = 3=1=(4-5), 3=0=(5-5), 3=0=(4-6), game force.
Opener bids 3NT with a double stopper or better in hearts;
With slam interest, opener bids 3S, 4C, or 4D showing his preferred suit, or 4H positive in both minors;
Without slam interest, opener bids 4S, 5C, or 5D showing his preference, or 4N to get partner to bid his best minor.
The minor suit distribution isn't terribly important, as long as you've got a hand with both minors, so partner can pick either one. It is important with these splinter bids to promise exactly 3 cards in the other major, so when pard has opened 1N with five cards in that major, you know you have the magic 8-card fit.