MSS continuations

This is mainly thought out for strong no trump range, such as 14-16 or 15-17. But I don't see any reason it shouldn't work with weaker ones when you know what you do.
This is a cramped auction and even though I could dedicate some more bids for minor hands, I don't think it's worth the cost as they are rather rare and everyone else faces same problems.
So all my minor oriented slam hands start with 2S over 1NT, MSS asking for better minor. 2NT is better D and 3C better clubs. Over both, 3m is a sign-off.
3M instead shows a shortness in that major but it tells nothing else about responder's hand type. He could still hold (13)(54), (12)55 or either long minor. It also isn't necessarily a slam invitational as you might be trying for 3NT in case partner has "waste" opposite your shortness.
3NT is a suggestion to play. It typically means that responder has a shortness in opener's better minor and a bad slam invitation. If as opener you happen to hold something like xxx xxx(x) in minors, you should definitely pull from 3NT. However 3NT could also be bid with a single suited hand without shortness that was slam worthy if opener fitted for his minor.
4C/D then are just single suited slam invites, not promising but not denying a shortness in other minor. How much the bid shows depends a bit on whether partner already showed a good fit for that suit. (Note that "good fit" might be just xxx, so it's not necessarily good)
4H/S These I reserve for 5 key exclusion bids (AKs of minors and major A) Mainly because it's a cool toy to have.
4NT is like stronger 3NT but it always shows shortness in partner's minor. It's quantitative, so not forcing.
5C/D shows a hand with a really long strong minor that was just too good to bid a direct 5m.

After 3M:
Opener's primary interest is to bid 3NT if there is waste opposite shortness. Any solid holding shouldn't be counted as waste obviously but it's a reason to downgrade. AQ isn't too bad holding either, so with it, you should only bid 3NT if you have bad support for minors.
Over 3H your situation is much nicer as you can bid 3S with all sorts of marginal hands. There responder can bid 3NT as nonserious. Over 3S you don't have such a luxury of waiting call before 3NT.
Over 3S when you don't have waste,  I think it's important to let partner know if you fit the other minor too because that might be the partner's real suit. So 4C tells that you don't have too special hand and no good support for the other minor (xxx at most) 4D tells that you have better than Qxx support for both minors but you are quite minimum. 4H tells that you don't have a support for the other minor but you hold a great hand otherwise. 4S then shows a great hand and support for both minors. Over 3H your 4C/D responses show great hands with no wastage respectively and 3S is any non negative.
At this point I think responder should know pretty well about the slam potential, so he can just bid next step as RKC for opener's better suit and second step for the other. If the response was 4S, 4NT is 6 key RKC. (I think it rarely hurts although it's not always optimal)

After 3M - 3NT:
When the responder has enough to go on even opposite the wastage, I think it's important to just clarify the hand type for opener. 4C/D shows the suit. 4H/S are cuebids with two-suiter (rebidding singleton here is just saying that nothing else to bid), 4NT is quantitative with (31)(54), 5m shows 11(65). (Bid the minor you know partner fits)

Couple general rules:
After responder has shown two-suiter, following RKC from either side should be for 6 keys (but not necessarily for 2 queens).
When responder has shown a shortness and one-suited hand, responder's bid in the shortness suit is RKC.