If you're hunting a fallen boss for the Rasputin Fallen Sword bounty, you have a few options available without having to tie yourself up in a strike.
Public events
Fallen Public Events, which take place on Earth and in Nessus, are a great place to find a Fallen Boss. They are generally quite quick to complete. If you do the Glow Extraction, you will need to make the event heroic by killing the little machines that spawn with each wave to spawn a boss at the end. If you do the Aether Supply with the Giant Servant Enemy, you can just kill him without making him heroic.
Ether Resupply will take place at Watcher's Grave and Exodus Black on Nessus, and Outskirts and Firebase Hades on EDZ.
Glow mining will take place in Trostland, The Sludge, The Gulch, and Firebase Hades on EDZ, and Tangle, Cistern and Glade of Echoes on Nessus.
If you want to use your time efficiently, you can try the new Seraph Tower event near the Winding Cove in the EDZ. This will throw waves of Fallen at you, including fallen bosses for you to kill.
Logistics K1 on the Moon
The K1 Logistic Lost Sector on the moon is home to a minion at the end who will also count as a Fallen Boss. This is unusual because these enemies are normally coded as a vehicle, but it will count towards this bounty.
Simply go to the area shown in the video above, then fight inside. You will need to clear each section before the energy shields drop and allow you to progress. When you get to the end, you'll find the Servant with a few boyfriends. Eliminate all the small enemies, then the Shanks that appear, and you can damage the Servant.
Take it out and the Fallen Sword bounty will be complete.
