Over the years there have been tons and tons of different bridge designs and style made by many different manufacturers. Some guitar manufacturers use tone pros bridges while others use a proprietary line of stop tails. Some guitar companies have designed to stick with original Floyd Rose bridges while others have decided to come up with their own Floyd Rose design. All of these bridges get adjusted in slightly different ways and hardly any of them actually use the same tools to adjust them. Some bridges use allen wrenches while others requires screwdrivers or custom tools to adjust. There is only one commonality amongst all these bridges: they all eventually need to be adjusted. That is where a guitar bridge tool kit comes in handy.
I know I titled this article "Guitar Bridge Screwdrivers", but as I mentioned above you really need an entire kit of tools to adjust different guitar bridges. I would suggest getting a number of different sets. That way you are prepared no matter what style of bridge someone brings into your guitar repair shop. The first bridge repair tools that I would get are two sets of allen wrenches: one American and one metric. Some bridge parts use American sizes while others use metric. It's a pain, but you have to have both sets. Also, some companies like Fender decided to put extremely small setscrews on some of their bridges. Some bridges require a 1 to 3mm allen wrench to adjust them. The second set of tools I would get is a full set of screwdrivers. You will need both flat-head screwdrivers as well as Philips. Usually these four sets of tools will allow you to adjust any style of guitar bridge. There will always be those weird custom tools that you have to get from time to time, but these will get you by.
You can't have a guitar repair bench without these tools. I don't even think you can own and play a guitar without these tools. You need some of these tools just to change the strings on a guitar. Plus, these tools are just good all around tools to have for other repairs. It sounds like quite a bit of tools to start out will, but trust me. You will need all of them. Personally, I like to buy Craftsman wrenches and screwdrivers because they have a lifetime warranty. You can buy whatever brand name screwdrivers or wrenches you like. Just make sure you get all the sets you need. If you would like more information about how to adjust your guitar bridge, please see my how to adjust your guitar bridge article.