By bidding 2NT partner presumably is saying pass if you're minimum else bid 3NT if your maximum.
So here, if you decide you're minimum you will pass.
If  you're a bit bolder you want to raise.  However, before you woodenly bid 3NT stop and think a bit.
Partner hasn't denied a 4 card major so it costs nothing for you to bid 3♠  on the way.
On this hand, partner will now bid 4♠ and you will be in the right contract.

What did you play to trick 2?
N was looking for a suit preference signal as he knows from the bidding you almost certainly have a minor suit Ace.
If you simply followed with your lowest heart, he will lead a club.
If you played the nine of hearts he will lead a diamond and the contract will now fail.

South's shape sounds like5-4-3-1 or possibly 5-4-2-2.
Either way S doesn't have a club stop.

11 pairs played in 3NT with only 4 making.  1 pair somehow reached 6NT.

I chose 4♠ which, given I've got two spade honours, means a 5-2 spade fit which hopefully should play well.

As indeed it does.

I don't know if some North's simply bid 3NT rather than 3♣ .

So what did you play from dummy at trick 1?

What actually happened.
Dummy played low as did East. Declarer won with the J♠ .
Whatever declarer now does he will be at least one off.  He tried to set up a club ruff but West won and led a spade, thus clearing the suit.
The contract is now minus 2.

At trck 1 dummy needs to cover the 8♠ . The 10 is the obvious card but the K works just as well.
The crucial point is the need to protect declarer's trump holding by forcing East either to win the trick or allow dummy to win it.

After this, with both red suit fimesses working, you'll emerge with +1 rather than -2.

S almost cetainly has 3 clubs and 3 hearts left.
N therefore has 3 spades to the Q and 3 hearts.

                         North
                       ♠ Q87
                       ♥ Q87

West                                    East
♠ KJ                                      ♠ 6
♥ J                                         ♥ AK95
♦ -                                         ♦ -
♣ KQ4                                 ♣ 7

                           South
                        ♥ 1032
                        ♣ J52

So, cash two club tricks and N must discard a spade and a heart.
Now cash the K♠ and S is squeezed and has to discard a heart.
Finally you can lead to the AK♥ knowing the Q♥ will drop and the 9♥ will be your thirteenth trick.

A satisfying double squeeze and that vital overtrick.

1 The play is irrelevant.  Have you and/or your partner paid attention to the bidding and thought to call the director?

2. To start with I'll assume you have a knowledge of 5 card puppet stayman.  Hence 3♣ asked N to describe his hand.  3♦ showed at least one four card major.
You see that South has both four card majors and must therefore know they have a four card fit in a major.  Hence South should bid 3♠ which shows a 4 card heart suit (but doesn't deny 4 spades).
N would bid 3NT to deny a 4 card heart suit.  From N's 3♦ bid S knows N must therefore have a 4 card spade suit and bids 4♠ .

Except that South passed 3NT.  Some mistake surely?

I hope you have read Gordon Rainsford's article on page 58 of the October 2018 edition of the English Bridge magazine entitled "Unauthorised Information".  If you haven't, I strongly recommend it.

Here, South has taken advantage of the unauthorised information that North has forgotten their system (or as was the actual case N/S hadn't agreed that 5 card puppet stayman applied in this auction).  Either way, S should have ploughed on beyond 3NT to 4♠ .
Both 3NT and 4♠ should make 10 tricks so the director should have adjusted the score from +430 to +420.

In case you think I've got this wrong, I've checked the hand and my reasoning with Gordon Rainsford who has agreed with it.

3. I leave you to think about the ethics of what would have happened if on seeing the lack of an alert by North to South's 3♣ bid, South hadn't alerted and simply bid 3NT.

4. Finally, if an opponent quotes a name of a conventional system they're using and you don't understand what it means, don't be afraid to ask for details.

What is partner's diamond holding?

Almost certainly a doubleton. In which case declarer has KJx.
It looks likely N has a heart honour and so declarer is likely to take a trump finesse.
If so, partner will win and can lead his last diamond.

So, did you  duck the first trick?
If you did, you'll hold 1♥ to seven tricks for a good score.
If you didn't, W will make an overtrick and you'll get a bad score (because 1NT by W just makes).

The double of 6♦ must be Lightner asking for an unusual lead.
Not doubling 6♥ doesn't want an unusual lead.

So, against 6♥ you should lead a spade.  Your reasoning should be that against 6♦ partner wanted a club lead so as to ruff it but against 6♥ she doesn't.  The bidding suggests a 4-4 diamond fit and a 5-3 heart fit.  If this is the case then partner must have five hearts and will come to a natural trump trick and try as declarer might, he must either lose a trump and a spade or two trump tricks. If you lead a club, declarer should duck in dummy and shouldn't have much difficulty in making 12 tricks.

At my table the bidding was as shown up to and including East's 6♦ .  S passed and N bid 6♠ which was doubled and went five off.

If partner can obtain the lead you want to keep all your spades.
If you don't think this is likely then should you hold onto your clubs or at least three of them in case partner has the K or Q?

I sugeest all this is irrelevant.  Declarer has bid hearts and is virtually certain to have the Ace and highly likely to hold the Jack.
The one card you mustn't discard is the 10♥ .

If you do, declarer will lay down dummy's A♥ , note you discard on it and take the now marked heart finesse.
If instead you follow to the first round of hearts, declarer will now regard a heart finesse as risky knowing that if it loses, you will cash the A♠ and whatever spades you have left.

The 10♥ may be of no use to you but it sure is of use to your partnership.

1. Most tables would have had this bidding and led their fouth highest spade.
Declarer now rapidly cashes 10 tricks.

2. Here East paused for thought and led the 2♣ reasoning that declarer couldn't have a four card club suit but could still have a four card spade suit.
The defence took the first five tricks.

You would like to lead from dummy to take the diamond finesse and lead towards your KQ of hearts.  Unfortunately you don't have sufficient entries and using your A♣ early on doesn't look the brightest of ideas. To concentrate your mind, your opponents are strong players.
You know E has four spades to the AJ but who has the nine?

What follows will not accord with the line recommended by the Play it Again feature as it cannot be regarded as scientific but deals with the impression you give to defenders. 
When a declarer attacks a suit it means he's looking to set up tricks in that suit, hence defenders should lead another suit.
When defending with dummy on your right, you look at dummy's weakness and expect to lead that suit unless there's a very good reason not to.

Here, declarer bore these two adages in mind and led a low club at trick two, covered East's 5 with his 8 and W won.  After a long think W led the 7♦ .  Put yourself in West's place; what would you have done?

This line of play could have gone horribly wrong if West had had lots of clubs but it proved to be worth trying.

When the J♦ won, declarer led the K♠ and was pleased to see the 9♠ appear from East.  This play couldn't really lose even if the nine hadn't appeared.
After that the play went - E took his A♠ and led a diamond won by declarer. A spade from declarer taking the finesse and cashing the fourth spade on which E discarded two hearts.  a heart from dummy taken by East's Ace.  East could have salvaged one trick now by leading a club to remove dummy's entry but instead chose to lead a diamond. Declarer now made the rest for the result of 1NT + 3.

With this type of auction, i you have an Ace and in the absence of any other information such as a bid from partner, it's usually best to lead it.
If you don't you might never make it.
Here East will make 13 tricks if you don't.

I am obliged to one of my readers for this suggestion as to how you should play and why.
His reasoning is that everyone will at least be in 5♣ and many are likely to be playing in 6♣ .
6♣ is cold whether you take the finesse or not.  Hence you cannot beat these pairs whatever you do.
Those declarers in 5♣ will most likely play with the odds and take the finesse. (Note: if N has all three clubs you will lose one club trick whatever you do) If you do likewise you can expect to get a joint bottom.  So, by playing the Ace you'll either get a complete bottom if the finesse was the correct play, or something like an average if, you drop a singleton King.

I leave you to decide.

Finally, would you have raised to 6♣ with West's hand?