
Two of the most extensive corkscrew collections I've ever seen are in Spain and Germany. In Spain, Dinastia Vivanco's wine museum has an entire exhibit room dedicated to corkscrews. In Germany, Von Hovel's tasting room is encrusted with them from floor to ceiling.
Artifacts at Pompeii and other sites confirm that corks were used to plug the holes in amphorae in Ancient Greece. Using cork stoppers to plug bottles evolved along with the glass-blowing industry. Dom Perignon experimented with stopping bottles with cork, and this practice became more popular starting in the 1790s, but this was still about 150 years before estate bottling became the "normal" way to sell wine. Nevertheless, ever since there have been corks, people have needed tools to get them out of the bottle.
In 1892 the crown cap became widely available. This closure never took off with wine, but it became the status quo for beers by the early 1900s. But corks were once a popular beer closure, and if you find old corkscrews from the mid-late 1800s, or even the early 1900s, they may have brewery advertisements on them. The idea was to create a high-quality corkscrew that people would use again and again, and each time they used it, they'd see the brand logo. In fact, Adolphus Busch (of Budweiser/Anheuser-Busch) practically used these corkscrews as business cards for decades.
While the crown cap overtook the cork in the realm of beer, corks remained popular in wine, and only recently have they been challenged with the screw cap.
Cork is made from the bark of the Quercus suber (cork tree), and corks are a renewable product. A cork tree can live up to 300 years, and the bark re-grows every ten years. I think it's interesting how the wine world has such direct ties to so many different species of oak. The tree genus Quercus (oak) has over 600 species. A little more than a century ago, you may have gone to any European dock and found a boat made from Quercus suber (cork oak)-- not the bark, but the hard wood trunk was used in ships because of its resistance to rot-- packed with barrels of Quercus robur (French oak) and Quercus petraea (Russian oak) filled with wine, and the barrels may have been stopped with corks made from the Quercus suber bark. Due to this complex inter-relationship between Quercus forestry and beverage production, an entire business of cork-extraction machines emerged over the last several centuries, and as a result, we have some inventive corkscrews in the world around us.
Here are a few photos of some interesting corkscrews:

Some are colorful.

Some have tines for extracting a cork from the sides.

Some act as spigots and have pour spouts.

Some are complicated.

Some are really big!

Some are funny.

Some are... even funnier!

OK, this one is just obscene.

Some are practical.

Others are utilitarian.

Some are advertisements.

Some are... a little weird.

And some are gold.
The Virtual Corkscrew Museum
Bull, Donald A. (1999) The Ultimate Corkscrew Book. Pennsylvania: Schiffer.
Bull, Donald A. (2009) Figural Corkscrews. Pennsylvania: Schiffer.