I know there is always stories centered on Noah's Ark, but I wondered if there were other ships historians were more focused on that has a much higher likelihood of actually being discovered. I believe there was a story around this time last year about Blackbeard's ship, Queen Anne's Revenge being found of the coast of North Carolina, but there must be countless other ships being searched for. I would love to keep track of those searches as they progress.
The go to source for news of archaeological work on shipwrecks is the IJNA. Here's a link to the journal. You'll need to have a subscription to get past the paywall unfortunately. A university library might also have a subscription or could get loaned copies. The Mariners Mirror is another major journal.
The Queen Anne's Revenge folks are pretty much sure of their find at this point and the site is somewhere over 60% excavated. Could be more I haven't talked to any of them in a while. They are not 100% sure its the QAR, and probably never will be, but they are maybe 90% sure and they aren't many serious objections left. Last I heard this was the final season of excavation as well and the active digging was winding down. Look's like their website got a long overdue overhaul as well
There has been an ongoing search for the Santa Maria, Columbus' flagship for a long time for obvious reasons. I saw a news article not long ago that a team think they found it. It's way to early to call it, and the with the ambiguities surrounding IDing sunken vessels they never know for sure. Here's the article anyways but don't get your hopes up. Like Dr. Beeker notes nothing is sure until an excavation is conducted. See above about the QAR where they have been working for almost 10 years and still cannot say with 100% certainty they are correct.
In the Great Lakes there is a long running effort to locate the Le Griffin. Le Griffin was a vessel built by French Explorer de La Salle and one of the first large vessels built in the Great Lakes. There are persistent claims that it has been found but none has been confirmed to date. The most recent is a large beam some say is a bowsprit from Le Griffin. Other say that testing has shown it to be much more recent. At this point there is nothing definite to be said really.
There is, or was, I haven't seen anything about it recently, an effort to find Henry Morgan's vessels. The media made a bit of a stir about it when Henry Morgan Rum backed the project. Here's a short article about it.
That's a few examples of high profile ships that are currently being looked for. You seem to be interested in ongoing projects so here are a few of those as well.
The Vasa is probably the most famous ship recovery ever. She sank as a result of a combination of under loading and poor ship handling. The harbor where she sank is extremely cold so preservation of the wreck was excellent. In 1961 she was recovered and a museum built around her. Conservation and study of the Vasa is ongoing. A few years back she was the subject of an extensive (84,000+) point total station survey. As the pride of the Swedish Museum system there is a 1000 year conservation plan.
The Mary Rose occupies a similar position in England. She was the pride of the fleet until she was lost in battle. Unfortunately she was sunk in warmer water and the sea scour and teredo worms had their way with her. Everything that was not buried was eventually destroyed. What was left was eventually discovered and now resides at the Mary Rose Museum. Because she tipped over after sinking one entire side has survived. There have been a number of documentaries and the double monograph on finds from the wreck came out not long ago.
The Viking Ship Museum in Roskild also has an excellent collection. They are the home to The five Skuldelev ships a series of Viking vessels recovered from Fjords in beautiful condition. They have been conserved and are now on display. The Viking Museum also has some of the best museum placement in the world because it sits on the waterfront. When the Danes went to expand the museum they chose to do so out over the water and when they went to place pilings they found MORE Viking ships UNDER the proposed museum. I think the last count was at 9 new ships.
These three projects also have a ton of articles in the IJNA for you to read if you're into the more technical aspects of their conservation, recovery, and discovery.
To give a slightly different example that /u/Vampire_Seraphin's a current excavation off of the coast of Sri Lanka at a fishing village called Gotavaya is the first marine excavation of an ancient shipwreck in Indian waters. It is dated from the first century BCE/CE, which was the peak of the ancient trade, but as of yet there has been no marine archaeology to help fill, out our picture, which is an enormous gap when one considers how much marine archaeology has added to our picture of trade in the Mediterranean.
The excavation has only just begun so there is no published material, but the team maintains a blog.