Unfortunately, most of them have not survived, although there are a few examples still preserved. The reason for that is that wooden ships rot, and they're difficult to maintain even when you have a skilled shipbuilding industry dedicated to them and their maintenance.

The most famous surviving ship from Trafalgar is HMS Victory, Nelson's flagship and currently the flagship of Britain's First Sea Lord, currently Admiral Tony Radakin. It's preserved in drydock in Portsmouth, and has been restored in various ways through the years -- it was laid down in 1759, launched in 1765, and served in various statuses until finally being moved to drydock in 1922. A substantial portion of its hull and fittings have been replaced over the years, and its "guns" are fiberglass (the ship was collapsing under its own weight). A current refit will involve major restoration of masts and rigging. Victory is the oldest warship still in commission.

The oldest commissioned warship still afloat is USS Constitution, which is docked in Boston harbor. It was laid down in 1794 and launched in 1797, and was in service until 1881, then served as a receiving ship until being designated a museum ship in 1907. By the 1920s the ship was in serious danger of sinking, having rotted and sagged extensively, and a combination of private and public funding eventually totaling almost $1 million (about $15 million in today's money) was needed to restore the ship, and about 85 percent of its original structure was replaced. It embarked on a goodwill tour in the 1930s to raise publicity for the Navy, and was restored again in the 1970s and again in the 1990s. Restoration efforts allowed Constitution to sail under her own power in 1997 to celebrate her bicentennial.

There have also been efforts to restore ships that sank in earlier periods --the Swedish ship Vasa and English ship Mary Rose are notable examples that have been raised and preserved. I wrote about both of them before.

To mention a ship that's near and dear to /u/itsallfolklore's heart -- if you're in Baltimore, you should check out USS Constellation, a sloop-of-war that was the last all-sail vessel built by the US Navy. Over time some controversy arose over whether that ship was the original Constellation, one of the original six frigates of the US Navy, and Folklore served on a commission to decide the truth value of that.

I'll just mention one more notable museum ship -- HMS Trincomalee, built in Bombay in 1817 by master shipbuilder Jamsetjee Bomanjee, is still afloat in Hartlepool, England as a museum ship. It was built out of teak, which is a fairly indestructible wood, and holds the distinction of being the oldest British warship still afloat.