VMF Kommuna. This 2,500 ton catamaran was built in the Netherlands and entered service in 1915. Kommuna began service in the Czar's navy, spent most of its career in the Soviet (communist) Navy, and now serves in the fleet of a democratic Russia. Originally designed to recover submarines that had sunk in shallow coastal waters, Kommuna remains in service to handle smaller submersibles, does it well and has been maintained over the decades to the point where it cheaper to keep the old girl operational, than to try and design and build a replacement.
The oldest ship still in service, in the world's major navy's, is the Russian salvage ship
Most navies would not want to bring attention to their oldest ship, especially if it was nearly a century old. It's different in the American Navy where, earlier this year, the carrier, USS Kitty Hawk (CV 63) was finally decommissioned, and ceased to be the oldest ship in the fleet. The Kitty Hawk served for 48 years and 13 days. In that time, about 100,000 sailors served on the ship. The ship was the navy's last non-nuclear carrier and, since 1998, the oldest ship in commission. "The Hawk" did not age well, and had lots of breakdowns in its final years. This led members of the crew to nickname the ship; "Shitty Hawk".
When the Hawk became the oldest navy ship in commission in 1998, it received the First Navy Jack (the Don't Tread on Me flag flown by the first navy warships.) The oldest ship in the navy is the only one that can fly the First Navy Jack, and that ship is now the USS Enterprise (which entered service seven months after the Kitty Hawk). Actually, since September 11, 2002, all U.S. Navy ships have been flying the First Navy Jack, and will continue to do so for the duration of the War On Terrorism. So it may be a while before only one ship flies the First Navy Jack.
The First Navy Jack moved around a lot in the 1990s, as the navy downsized because the Soviet navy had largely disappeared, including some ships nearly as old as the Kommuna. The USS Prairie, the last of the pre-war US Navy ships that fought in World War II was decommissioned on 27 March 1993. The ship, a destroyer tender (a supply and maintenance ship for deployed destroyers) entered service in late 1939. The USS Prairie passed the First Navy Jack on to the USS Orion, a submarine tender commissioned in 1944. But the Orion went out of service later that year and the First Navy Jack rapidly moved from one retiring ship to another until the Kitty Hawk got it in 1998, and held on to it for eleven years.
Technically, the oldest ship in navy that is still in commission, is the sail powered frigate the USS Constitution. But this is a memorial and museum ship and the "commissioned" status is basically honorary. The two century old Constitution can no longer sail, or fight. Kommuna and Enterprise can.