Images are provided on the right, save for the First and Sixth Doctor's ones, which only appeared in non-visual media.
- First Doctor: Occasionally used one in Virgin Past Doctor Adventures novels.
- Second Doctor: Original design, the one that looked like a penlight. First seen in "Fury from the Deep", and last seen in "The War Games".
- Third Doctor: Classic prototype design, the one with the pull-down cover and replaceable emitter head. First seen in "Inferno", and last seen in "Carnival of Monsters".
- Fourth Doctor: Classic design.
- Fifth Doctor: Classic design, until "The Visitation", when it broke and he didn't replace it.
- Sixth Doctor: Occasionally used one in Big Finish audios.
- Seventh Doctor: None until "Doctor Who": the Movie, then the Movie design, with the telescoping end and red bullet-shaped crystal.
- Eighth Doctor: Movie design. First seen in "Doctor Who": the Movie, and last seen in "The Night of the Doctor".
- War Doctor: TIme War design. Only seen in "The Day of the Doctor".
- Ninth Doctor: Blue design, the fancy one that glows blue.
- Tenth Doctor: Blue design.
- Eleventh Doctor: Blue design in "The Eleventh Hour", then multiple different versions of the Green design, the big copper one that glows green.
- Twelfth Doctor: Green design until "The Witch's Familiar", then Blue-2 design, the blue time-rotor-shaped one, from "Hell Bent" until it was destroyed by a smartsuit in "Oxygen". He then replaced it with a Blue-3 design, which was a replica of the previous one, used by the Virtual Doctor in "Extremis", and the "real" Doctor from "The Pyramid at the End of the World" onwards.
For some of these designs, there's more than one screwdriver. For example, Ten broke his at least twice and replaced it with an identical model, and Eleven broke his at least once and gave it away at least twice. There may well be more replacements we don't know about. On top of that, while Three and Ten both liked to tinker with their screwdrivers, we can't really be sure that all of the minor variations were just tinkering since last week instead of a whole new screwdriver.
Out-of-universe, there was generally a single prop for each screwdriver for each design, but the original blue prop was replaced by a pair of slightly different ones (explaining some of the variation between episodes), and the Eighth/War prop was actually a modified Fourth Doctor toy because the original prop was long gone. The Classic and Blue screwdrivers were also modified, repaired, replaced, and repainted quite a bit over the years.
The audios, novels, and comics make things a lot more complicated. For example, the Eighth Doctor reached back through time to steal his original screwdriver from the Second Doctor, used it for a while, and then later gave it to the First Doctor, making it a causal loop (also known as a Bootstrap Paradox. Click here to see the Twelfth Doctor explain this concept).
However, as the Moment said in "The Day of the Doctor", "It's the same screwdriver. Same software, different case."
If you want all the gory details, see Sonic Screwdriver at the TARDIS Index File.