3
♠ for me, and I don't think its close between that and x. Refuting Bid-em's points:
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1) The scoring is IMP's, I really don't have to play in a major suit to do ok on the board. We should attempt to find our best fit. Bidding 3S or 3N directly, does not involve partner in this decision. In addition to having spades, I also have a fit with either minor, so lets try to find out what partners hand really is, instead of insisting on spades or NT with our first bid.
True, but 10 tricks are cheaper than 11, and we can't be ecstatic about it when pard bids 4 minor.
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2) We can be reasonably certain that partner is not leaving the double in. If he does, we are still quite happy.
Possible, but doubtful with our A-J. Unless RHO really preempted on trash, you aren't getting to play 3
♥ doubled.
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3) We can be reasonably certain that partner cannot bid 3N on his own. Playing 3N only being able to hold up once may not be such a great idea (although I prefer the choice of 3N to 3S(......and)4) If partner can bid 3N by some odd miracle, we are quite content. Not only that, we are better placed, since an opening heart lead lead will be going into his hand, and not thru it. And any non-heart lead by RHO may well give us the tempo required to make 3N.
Similar to the penalty pass argument, its very unlikely pard can bid 3N. But he's just as apt to bid 3N over a double than 3
♠.
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5) It is quite likely that if I can indeed make 3N, that 5 of minor is safer, especially from the other side of the table.
This I don't understand, but what does this have to do with the double vs 3
♠ argument? If we make 3N its on a source of tricks, and its very likely that we have too many losers for 5 of a minor. I'd be curious in a reasonable construction where pard can bid 3N, but 5 of a minor is better. I think it requires us to be totally off one of the minors.
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6) Partner may well have spades and bid them on their own.
Great, then we'll find our slam when I overcall in my AK-5th.
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7) Partner just as easily could have a stiff or doubleton spade and a 5 or 6 card minor, where 4/5 of minor is cold, but 3S/4S goes down.
This is the best argument for double I think, especially against playing 3
♠ exactly. Its very possible that pard has a very weak hand with a long minor where even 5 of a minor is better than 3
♠.
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8) It is unlikely that we will ever find the minor fit, if we just bid 3S or 3N.
Its also very unlikely we will find spades if we double.
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9) Partner probably will not take us off of 3S on a holding such as: xx xx KQxx AQxxx (maybe he should, but many wouldn't). Note on this holding we probably only make 3S, but 5C is certainly reasonable. Partner may also raise us to an unmakeable 4S on this holding.
(and....) 10) Good partners bid 3S on 3-3-4-3 holdings and 3-2-4-4 holdings when weak.

Keep that in mind when you are dealt a 3=1-5-4 and want to double 3
♥.
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11) Experience has proven to me that X is the best call. It goes against everything that one is ever taught about bidding in these situations. I fought this for years with my current regular partner. The results have proven (to me) that doubling has to be best and it is practically automatic now. And it works.
When I hear comments like this I tend to think that either you haven't had a representative set of hands in your experience, or that you have a selective memory and ignore the cases where you have missed the boat on a perfectly mundane 5-3 spade fit.