Up to date solution as of January 2021
Challenges appear on the Director as a gold medal icon
When you hover over the vendor or activity that's offering a challenge, you'll see it as a gold bar on their popup. This screenshot shows two challenges for Nightfall: The Ordeal
New challenges appear at the weekly reset and you'll have until the next weekly reset to complete them. There appears to be no way to view how many challenges you've completed, so keep track of it yourself.
You may need to do more than 30 challenges; it didn't pop for me until I did 37. My 37th challenge was for completing Crucible bounties, and it popped after I talked to Shaxx and claimed my reward. You may need to do the switch profile trick.
The exact challenges offered can vary week-to-week, but the following is a list of frequently-appearing challenges, what vendor/activity offers them, and how to complete them. I've read that some of these challenges won't appear until you've hit the soft power level cap, which is currently 1200.Zavala: Complete 8 Vanguard Bounties
Zavala's bounties are for Strikes, and require a number of kills with a specific ability or weapon type or of a specific type of enemy.Gunsmith: Complete 8 Gunsmith Bounties
The Gunsmith's bounties are framed as "calibration," and are for kills in a specific world, activity, or with a specific weapon type.Shaxx: Complete 8 Crucible Bounties
Lord Shaxx's bounties are for completing or winning Crucible matches, sometimes in specific playlists, or for specific categories of kills.Drifter: Complete 8 Gambit Bounties
The Drifter's bounties are for winning Gambit matches, banking motes, or for doing specific things during matches.Hawthorne: Complete clan bounties
Suraya Hawthorne's challenge is actually
for earning 5000 Clan XP. She does offer bounties for doing activities with clan members that will give you Clan XP, but those aren't required. Simply being a member of a clan and doing activities solo will earn you Clan XP. This older Bungie.net post
seems to line up with my experience of what activities give Clan XP and how much. You'll very likely knock this out without even trying if you're working on bounties for the other vendors.Vanguard Strikes: Complete 3 strikes using the same subclass element as the currently active elemental burn
When you tap on the playlist, there'll be a set of icons off to the left. Hover over each one; one of them will be an elemental "Singe." Equip the subclass for that element and complete three strikes in the playlist.Complete Nightfall: The Ordeal activities. Higher difficulties grant the most efficient progress
If you're playing on the lowest difficulty, it will take three strikes to complete this challenge. I don't know how many you need on higher difficulties.Complete Nightfall: The Ordeal with a team score above 100,000
You have to play on Legend or above for a score that high to be possible. I've also seen some older articles that say you start losing points after 20 minutes, so completing the strike quickly is important.
Crucible and Gambit usually also have challenges for completing a few matches.