The Salarian Ark
Instead of finding the Salarian Ark via hunting down leads or pursuing side missions, the ship rather conveniently shows up unexpectedly as part of the main story itself.
For the final stretch of the mission Hunting the Archon, which takes place about ten hours into the story, you’ll be assigned with locating a Kett flagship in the Tafeno system. Travelling to these coordinates, however, will bring you face to face with the Salarian Ark instead, which has been captured by the Kett.
Board the Ark to start the mission - what follows is a lengthy, linear quest to hunt down the Archon, figure out what’s happened, and save the Salarian people. This sequence features a lot of combat, including multiple boss battles, so don’t venture in without the proper combat equipment.
After exploring for a bit you’ll find the Salarian pathfinder, who will explain that her crew are being held hostage on the Kett’s ship, which has tethered itself to the Ark.
From here, you’ll head over to the tethered Kett ship to take out its crew, free the Salarian hostages, and sever the Archon’s grip over the Ark.
At the end of the mission, you’ll be given a choice to save Salarian pathfinder Raeka or save the Krogan scouts being held captive by the Archon, but the Ark will be restored and its passengers saved regardless of your decision.
After completing Hunting the Archon, you can head over to the Nexus to find the Salarian Ark has made it back in one piece, thanks to you.
Congratulations Pathfinder - you've found all three missing Arks.