Here is what I found as this current week is Mercury(S10 Wk2). Brother Vance Bounties = 0% Mercury Patrols = 0% Regular and Elite Mercury Defenders and Conquerors = 0% Elite Defenders and Conquerors are High-Value Targets. Regular Adventures = 0% Storm Minotaurs from Vex Rings = 0% Cabal Prisoners = 5% Must kill all 3 Psions and the Centurion. Terminus Keeper Hydra = 5% Spawns from unlocking the lost treasure chest and you only have about 2-3 minutes to kill it. Lost Sector = 10% Current Heroic Adventure = 20% Regular and Heroic Public Event = 20% Both versions give the same percentage but Heroic should definitely give 10% more because of its low spawn rate. The quickest way in my opinion is to run the lost sector until the PE spawns then just burn through it and repeat until the Flashpoint is done.

The lost sector and heroic adventures. Pretty sure patrols don't count(unless they changed this) and the sundial doesn't count.

I did Lost Sector twice, as well as two public events. One or two sets of Vance's bounties, some killing around the area (ok, a lot of killing). Opened a last remaining region chest on one of the islands. Y'know, did various stuff, didn't exactly followed what counted towards the progress. I looked once at 70%, second time it was only 3-4% short. It happened eventually.

Patrols and Lost Sectors advance flashpoints by 5% each. The Sundial does not progress your flashpoint. Like DC said, you might want to clear your cache.

Try clearing console cache & retry. I was getting progress in lost sector. Also, I don’t believe the sundial activity counts as that’s a separate activity.