I just watched the episode The Almost People last night, and toward the end, the real Doctor threw his sonic screwdriver to the ganger, who then proceed to destroy the monster and himself in the process. I assumed the screwdriver was destroyed with the castle. Once aboard the Tardis though, the Doctor has a sonic screw driver and uses it on Amy. I've always assumed he only had one, partly because of the episode with the planet wide library, he had given River the screwdriver prior to her showing up there. He seemed shocked he would have parted with it. Does he have more than one?
At the beginning of Series 5, we see the TARDIS Console provide him with a new one (the green-lit model he's been using ever since). Presumably, somewhere along the line, he had a sonic screwdriver dispenser fit onto the console :-D
In "The Day of the Doctor" it is stated that it is the same screwdriver, although in a different case, which allows the Doctor to perform a long (centuries long) calculation in a very short time (from the viewers perspective) as the 3 Doctors (war Doctor, 10th and 11th) are all in the same room:
MOMENT: It's the same screwdriver. Same software, different case.
WARRIOR: Four hundred years.
DOCTOR 10: I'm sorry?
WARRIOR: At a software level, they're all the same device, aren't they. Same software, different case.
This implies that at the heart of it the software (i.e. the part that makes the screwdriver function) is the same, but the outer case is essentially disposable and simply the interface the Doctor uses.
While David Tennant was playing the Doctor he destroys his sonic screwdriver, saving Martha and at the end of the episode he has a new one. So I think he could have many back-ups in the Tardis.
I don't know about Uncle Mikey's answer but the Doctor is a Time Lord with a Time Machine. I would assume that he would go to a time period and find another sonic screw driver. Or make one.
The Doctor has been using a Sonic Screwdriver since the Troughton era. While its design changed slightly over the years of the original series, it's always referred to as "the" sonic screwdriver, implying there's only the one. Any changes to its looks could conceivably be ignored - while the prop may have changed, you could argue that it's still the "same" device in continuity. But there was a different prop in two, three, and four/five's eras. The one in the Troughton episode was just a pen flashlight.
The screwdriver was destroyed in the Davison years, and was not replaced for the rest of the original series. This was due to John Nathan-Turner believing that it had become a "magic wand" and was used too often to get The Doctor out of too many jams ( an argument made by quite a few fans today, but I digress)
The War Doctor had one again, one that looked very much like the "classic" design, with a number of modifications, most notably the emitter beam head was gone.
Eccleston and Tennant had the same model, differing only by color. As mentioned already, Tennant's was destroyed in Smith and Jones, and either replaced or completely repaired.
A modified version of that model was given to River Song at Darillium.
The screwdriver was destroyed again in Matt Smith's first episode, replaced by a brand new model by episode's end. Capaldi used that model for a series of three, eventually getting his fancy new one.
The official Doctor Who Sonic Screwdriver App has eleven devices. It offers each of Two, Three and Four/fives's devices. It also includes The Doctor's Sonic Sunglasses, and The Master's Laser Screwdriver, but only listing one for Nine and Ten, ignoring the color change
So if you want to discuss discrete models used by The Doctor himself, there's eight, if you count nine and ten's as different. There's been a couple replacements of a particular model in among there.
But as already's been mentioned, they all use the same basic software, so you could argue they're all the "same" device, only with everything else about it changed. Much like the person who wields it.