If you're going to go, go. If you're going to play employers to get more compensation, play prospective employers.
If you take a counter-offer from your current employer, they'll just put you on the shit list. First to go in any sort of rough terrain. First to get thrown under the bus if they need a body. And most importantly: you'll have set a new normal for salary discussions. That is, they'll feel they can simply say 'no' to every request until you're ready to leave again and then just match the offer.
They get to continue paying you X for however many months you give them; however many months it takes to find a job worth leaving for. Then they coast in and match someone else's 1.2X offer, having enjoyed months and months of paying your old salary and playing all that stress and frustration and work of getting another offer against you.
And you can be certain they'll be playing every social/moral game to make you think you shouldn't get it along the way .
Further, you'll have damaged the nascent relationship with the would-be hiring company. They invested nontrivial time and energy vetting and planning for you and will be miffed to find they were a pawn in a raise-play. Sure, it's a valid business decision to change your mind and turn down an offer. But those individuals will remember and give you less preference in the future . Taking a counter offer from your current employer just isn't seen the same way as taking a competing offer from a third firm.
By all means, use a counter-offer to negotiate for more from would-be new employers . But don't take the counter offer and stay put. Unless you really love searching for jobs and can dispassionately go through the "I'm going to quit." stage and that doesn't negatively impact your co-workers .
 "it's not a good time right now". "the economy has everyone really tightening down". "no-one's getting a raise this year". "i had to really push to get you guys a bonus, there's nothing left for salary". etc.
 They'll also see you as one of those "threatening to quit to get the best possible raise" people. Which is a negative for any manager who'd rather not go through that drama.
 Keep in mind that asking for a bid against a counter-offer can blow up if you have no intention of taking the counter. If you push for them to bid and they decline, accepting that job marks you as a certain type of negotiator, which isn't helpful for you. So if you really want to take this route, be prepared for the case where you need to actually accept the counter-offer, but continue your job search anyway.
 Remember that in any group-work situations, your co-workers will catch nontrivial shit when you move into the negotiating/quitting stage and they have to adjust and plan around your possibly not being there next month. Which they'll be forced to do, because management will ensure they can most-effectively bargain against you.