Picking up from where we last spoke on the subject (check it out).
I suspect that some form of Puppet Stayman is the most popular system nowadays (2021) among experienced players. As with any convention, discussion of follow-ups is very important. Any bidding structure over 2NT (which swallows up a lot of important bidding space) should be able to solve the following problems:
My suggestion here is perhaps slightly less efficient than the best versions of Puppet Stayman, but it is much simpler and easier to manage. It must be remembered that this is a low-frequency issue; any budding partnership should devote their time to more pressing matters before trying to fine-tune this sequence, which is why a simple alternative is often best.
Here it is:
After 3C, opener bids:
3D: denies hearts. Can still have spades (but can also have no major)
3H: 4+ hearts. Denies 4 spades.
3S: 5 spades
3NT: both majors
Over 3D from opener, responder, if interested, asks with 3H. Opener bids 3S with 3 cards in spades (right-siding the hand when responder has a Smolen-type, 5S4H), 3NT with 2 spades, and something at the 4-level with 4 spades.
If responder bids 3S over 3D, he is showing 4 spades and 5 hearts. Notice how responder bids the Smolen hands just as he used to do with plain Stayman; he bids the 4-card suit after a 3D response.
After opener bids 3H, showing hearts and denying spades, responder can (if he feels like it) inquire about whether opener has 4 or 5 hearts by bidding 3S. Opener will only bypass 3NT when he has 5 hearts. Note that this also establishes hearts as the trump suit for slam investigation, if responder bids something forward-going in the next round. (Again, this is very much like expert bidding over plain, 70's style Stayman).
The other 2 responses to the 3C inquiry are easy to deal with. If opener bids 3S, showing 5 spades, responder bids 4H to establish spades as the trump suit for slam. Other bids are natural. After 3NT showing both majors, it is useful to play (at least) 4Red as a transfer to the next suit, to allow for setting the suit, right-siding the contract, and exploring for slam.
So there it is. An alternative to Puppet Stayman which is much easier to navigate than the most exotic varieties out there.