Declarer started with seven diamonds.
Two club tricks have gone so you will have to come down to four cards.
So this looks like K♥ , Q♣ and 2 spades.  Keeping 3 spades to the nine hardly looks sensible.
Now count declarer's remaining 4 tricks.
He must have the A♠  and will make the A♥ and the K♠ .
If he has the Q♥ then he will make the J♥ for his 12th trick.  Hence you must assume partner has the Q♥ .
What about clubs?  If you've been counting, there's only one left and partner's play to the first two tricks indicates he started with three.
(He will also have discarded it on one of the diamond tricks.)
Hence your Q♣ and the H♥  are worthless!

So, despite it looking unlikely, you must keep your three spades.  (Partner had come down to Q,J ♠ and Q,7♥ )
Lo and behold the A,K,Q,J,10♠ all fall on the 11th and 12th tricks so your 9♠ is the setting trick.

Ten tricks looks straightforward provided you can avoid losing a trump trick.
Did you stop and think before playing from dummy?

If you didn't, you probably saw North win with the King and you followed suit with the 6.
Oh dear. Now North cashed the A♥ and led another.
Whatever you now did meant you were doomed to be one off.

Before playing from dummy, you need to think why South has led the suit bid by your partner?
It should occur to you that this must be likely to be a singleton heart.
If North cashes his second heart and leads a third you will have to ruff with the J♠ and hope South doesn't hold the Queen.

Could you do any better?
Yes you can.  North may still defend in the above way.
However, put yourself in his position if you were to play the Q♥ to trick 1 .
He now can't be certain whether you or his partner has the singleton.  Hence he might not want to set up the J♥ and so switch to another suit.

An important point to note.  You must think this out before playing from dummy.  If you stop and think after North played the K♥ then you've blown it.
North will immediately know you must have more than a singleton, otherwise what are you thinking about?
Only an unethical player would take time to think before playing a singleton and you're not one of those are you!

From South's double and having opened the bidding, it looks as though she must have the Ace of diamonds and probably the King of clubs.
Hence your first thoughts are to take the club finesse and later lead twice towards the KQ of diamonds.

The problem is you only have the Ace of hearts as an entry to your hand until trumps have been drawn.
Hence you can't afford to play hearts at trick 2 because even if you pick up trumps for no loss, you will then have to lose to the Ace of diamonds after which 
defenders will cash two heart tricks.
So you you have two choices
a) Cash the Ace of clubs and hope for a singleton King
b) Lead the King of diamonds hoping South will take her Ace.  If she does, you can now come to hand with the heart Ace and take the club finesse. 
You will still need to consider taking the finesse of the Jack of diamonds.
If she doesn't take her Ace of diamonds then you will have to cash the Ace of clubs at trick 3 and hope.

As it happens both lines of play work, assuming you finesse South for the Jack of diamonds if you try b).
Personally, I think line b) is best but it's the initial thinking on how to play the hand which you should consider.

If you bid 3 clubs South will bid 3 diamonds which will be passed round to you.
You then have to decide whether to Pass or bid 4 clubs.

Here a UCB (Unassuming cue bid) works much better.  Bid 2 diamonds.  This shows good trump support and 10-11 points.
West can now bid 3NT reckoning East must have the Ace of clubs and must be likely to contribute one other trick.

My thanks to Helen Erichsen for this analysis.

It looks unlikely that others will be in 4♠ but what will be a good score? Not yet knowing the position of the two minor suit Kings, it's impossible to say.
It does look as though the diamond finesse will have to work.
You play low from dummy and West played the 8♥ .
Did you ruff or discard?

How are the trumps going to break?
If they are 3-3 then you can afford to trump.  If they are 5-1 or 6-0 then you have no chance. What about the most likely 4-2 split?
A key part of maximising your tricks is not to lose control of trumps.  Hence you should discard.
West would be foolish to cash the A♥ and so has to switch, probably to a trump, thus enabling you to keep a step ahead.

As the cards lie you will now make 10 tricks for a top.

Ah, but what did I do I hear you ask? 
Obviously I ruffed trick 1!

Note also that East could have ensured my defeat by leading the J♥ .

Rather than bidding 5 spades, it can't cost East to bid 5 hearts instead.
West will convert to 5 spades and will now know what to lead against a club contract.

Several pairs got to 6NT.
Scientific or a punt?  I don't know.

Re responding to an opening 4NT, 5 clubs would show no Aces and 5 diamonds would show one Ace which is specifically the diamond Ace.

Partner must have a strong hand and be at least 5-5 in hearts and diamonds.

Now, what is your partnership agreement on showing Kings after 5NT?
If you show your number of Kings then you will bid 6♦ to show one (ignoring the King of trumps as you've already shown this)
If you show specific Kings then you will also bid 6♦  .  If you had no Kings (apart from trumps) you would simply bid 6♥ .

Here, partner can bid 7♥ with ease if she knows I have the King of diamonds.  But if she only knows I have one King then which King is it?  Oh dear, 7♥ was not a success.

My suggestion is that agreeing to show specific Kings will be better for you on more than 50% of occasions.

By playing the Q♠ partner is promising the J♠  (or he has a singleton)
Hence you should lead a small spade at trick 2.
If you lead the K♠ , declarer can set up the 10♠ for an extra trick.

The general wisdom is that if you go past a making 3NT then you might just as well bid 6 of a minor rather than stop in 5 as 5 making and 6 minus 1 will both be poor scores.

However, on this hand I wasn't sure that at other tables many ,if any, N/S's would play in 3NT..

Now, considering 6♣ , unless S has AKQ of spades we are virtually certain to lose a spade trick.  The diamond lead is marked. So, unless S has a void in diamonds we will lose a trick in that suit as well.
Hence I settled for 5♣ .
As it turned our S was void in diamonds so six was made.  Only two pairs bid six and none played in 3NT.  One poor North did play in NT but ended up in 6.  -1 was a bottom.

The only hope is to eliminate the spade suit, draw trumps and throw South in.

After the last spae is ruffed by West this will be the situation.

                                                   N
                                          ♥ Q 10 9
                                          ♣ J 10 7

    ♥ 7 5 2                                                  ♥ K 5
W ♦ K                                                       ♦ J 8   E
    ♣ K 8                                                    ♣ 53

                                                S
                                           ♠ A
                                           ♥ A J 
                                           ♣ Q 9

West now leads the K♣ .  Now South has to be aware of the threat of a throw in and must 
discard the Q♣ .
Would you as South have spotted this?

The crucial question is whether or not to take the trump finesse.
South's switch to the three of clubs suggests a singleton.  However, looking at dummy, it appears that South would be likely to switch to a club and the two of clubs is still missing.
There's no right answer.  As it happens the winning line is to cash the Ace of trumps and lead another spade.  South now cannot ruff a club.

Looking at all four hands, can you see how N/S can defeat 4 spades regardless of how declarer plays?

What can East have for this vulnerable 3♦ overcall?
From the lead she almost has a singleton club.
7 or 8 diamonds?
Your only hope is that East has QJx of hearts and that you can eliminate her black cards. force her to trump and then have to lead a diamond.lead.
For this to work, East must have at least two spades.
So cash the King and then the Ace of hearts followed by the jack and Ace of spades
Now lead the King of spades.  As it happens East has the 10 but after the next lead of the King of spades East is stuck.

3♦ doubled would have given us +1800.
Oh well, +980 scored well.

I assume you are using Michaels cue bids so 2♦ and 2♥ are all possibe.
You don't have enough points to double.
1♠ is also a possibility but this isn't likely to upset N/S's bidding.

I suggest this is the ideal opportunity to use the Unusual 2 NT.
I suspect most players automatically associate this bid with having both minors.  Here 2NT can't be 20-22 pts so it must be unusual.  It can only be unusual by having the two black suits.

After 2NT South bid 4♥ .

Now look at West's hand.  Thi, frankly, is lousy.  However, the vulnerability is favourable so 5♣ dblrd can afford to go 3 off as 4♥ is virtually certain to make.
West counts likely losers in clubs: 3 in the red suits, 3 or four probably in the black suits.
So there is a reasonable chance that 5♣ dbled will only be -3.

Further, bidding 5♣ gives N/S a guess.  Should they double or bid onto 5♥ ?
This added consideration confirmed West's choice of 5♣ .

Win the K♦ .
What you mustn't do is to lead a trump at trick 2 as N will play the Ace and lead a trump.
Instead, cash the AK♣ , the AK♠ , fuff a spade, cash the A♦ discarding a spade.
Now East only has trumps left.  Lead dummy's last trump .......

You should bid 5 spades.  
But I haven't got the Queen of trumps I hear you say!
Partner has twice shown he has five spades; first when he bids 3 clubs and then when he bids 3 spades.
Hence you know you have 10 spades between you and so only a 3-0 trump break might cause you problems.

Let's revisit the auction and see how it can be improved.

 N     S
1♠    2NT
3♣    3♥ 
3♠     4♦
4NT   5♠ 
6♠     Pass

4♦  is preferable to 4♥ as this shows second round control in diamonds which is likely to be much more useful information to North than 4♥ .

Either of four hearts or double would have done.

Four hearts can be defeated but this requires a double dummy defence.  Three spades is likely to fail by two tricks- this depends on the lead and how declarer plans the play.
South should reason that -140, +50 or even +100 are unlikely to be good scores so one of double or four hearts needs to be chosen.

A jump to 3♠ looks to value the hand.  If partner is minimum he will pass, otherwise he will bid 4♠ .
But what if partner has a good hand?
Here's Espen's explanation.

2♦ would be forcing and showing 16+ points.
3♦ would be a splinter, agreeing spades, showing a good hand with a singleton diamond.
4♦ would be a splinter, agreeing spades, showing a good hand and a diamond void.

Hence here you should bid 3♦ .  Partner should get excited and you will likely end in 6♠ .

One West bid 5♣ .  The bidding continued 

  N     E      S        W
5♠      P      P        6♣ 
6♠     Dble  P       P
P
Contract made

My observation on this is that West having felt his hand was only worth 5♣ should then pass.

At the other table  West bid 6♣ .  The bidding continued

  N     E      S        W
6♠      P      P        7♣ 
7♠     Dble  P       P
P
Contract -1

My observation here about West's club rebid is different.  He would hardly go to seven initially so bidding one more is not unreasonable.
East double I suggest is unwise.  Yes he knows 7♠ is bound to go down but would 7NT?
I wonder if North thought of bidding 7NT?

Every West continued with a diamond presumably reckoning that partner held the AJ over the Q.

The lesson here should be to stop and count.  The 3♦ bid should be showing a 7 card suit.  Counting the suit indicates declarer's seven is a singleton.
Hence in a) and b) you should switch to the A♥ and then lead another heart.

I decided that most E/Ws would be in 6 hearts in which case +1300 would be just as much a bottom as going 1 down in 7 hearts.
So I bid 7 hearts.
S led the 7 of spades which E read as a singleton and with trumps dividing 2-2 she could draw them ending in dummy take the marked ruffing finesse in spades, cash the A♦ , ruff a diamond and run the spades to claim 13 tricks.

On seeing the other scores I was interested to see that my analysis was wrong.  +1300 would have been a top as only 2 pairs had bid 6♥ and neither made it.  The only pair to make 12 tricks had stopped in 4♥ .