From Destinypedia, the Destiny wiki
Various Events are available as high-level endgame activities designed to be played after reaching Level 20. The rewards can include high-end loot, but are only available after a single completion with each character. 
These events are always available at a regular basis.
Daily Crucible Event
Every day one Crucible gametype is chosen at random to be that day's Crucible event. This can include one of the rotating playlists not in regular circulation. The daily event offers more rewards than are normally available from a single match, including Passage Coins to get into the Trials of Osiris.
Weekly Crucible Playlist
Each week a Crucible playlist is chosen at random and featured for the entire week. Similar to the Vanguard Heroic Strikes, players can earn rewards for the first three matches completed in the playlist each week. Rewards are account-wide, rather than per character; completing three matches with one character and switching to another character will not grant any further rewards until the weekly reset.
Daily Heroic Story
Every day a random Story Mission is selected. The mission features the Heroic modifier, and offers more rewards than are normally available from replaying a Story mission. A minimum Light level of 350 (formerly 240) is required.
Certain Story missions have alternate endings which can only be accessed when that mission is selected for the Daily Heroic Story. Known missions with these endings include:
Weekly Heroic Strike (Year One)
Each week, a random Strike was selected for the Weekly Heroic Strike. This Strike had both the Heroic modifier and an additional random modifier. Selecting a higher difficulty increased the reward payoff. Matchmaking was allowed.
Vanguard Heroic Strikes (Year Two)
Vanguard Heroic Strikes replace the Weekly Heroic Strike. Players can play the Vanguard Heroic Strike playlist and earn rewards for the first three Strikes completed in it each week. Rewards are account-wide, rather than per character; achieving three wins with one character and switching to another character will not grant any further rewards until the weekly reset. All Strikes in the Vanguard Heroic playlist feature both the Heroic modifier and three random modifiers. A minimum Light level of 260 is required.
The Nightfall Strike is an extremely difficult Level 42 strike with five modifiers total. One of the modifiers is Epic, which replaces Heroic. The remaining four modifiers are random. Matchmaking is unavailable. The Nightfall is always a different Strike from the Weekly Heroic Strike; before House of Wolves, the two used the same Strike each week. A minimum Light level of 380 is recommended.
In Destiny 2, there are two kinds of Nightfall Strike; The Ordeal, and regular Nightfalls. One Strike is chosen to be The Ordeal and three to be regular Nightfalls each week. The latter has a recommended minimum Light level of 820 and will disable equipment changes; modifiers can be applied by the Fireteam leader for a score bonus using the Five of Swords item, acquired from Xûr, Agent of the Nine free of charge. The Ordeal features five difficulty levels, the easiest having a recommended minimum Light level of 750, with the rest increasing with each season. Each difficulty features more modifiers and better rewards than the last.
In Beyond Light, regular Nightfalls were removed, leaving only Nightfall: The Ordeal.
These events are only available on rare occasions or under special circumstances.
Festival of the Lost
For one week, Petra Venj appears in the Tower as the Queen's Wrath, offering Bounties for wanted Fallen found on Patrol. A new Bounty becomes available each day. Hunting down these enemies gives players the chance to open an Ether Chest, and buy Emblems and Shaders.
Sparrow Racing League
Trials of Osiris
Every week, from Friday to the weekly reset on Tuesday 2 AM Pacific, the Trials of Osiris becomes available to those who own the House of Wolves expansion. Like the Iron Banner, it's a high-challenge event, but what sets it apart is that entry must be paid for. See article for details.
Solstice of Heroes
The Solstice of Heroes was available to every Destiny 2 player during the entire month of August 2017, and was made to celebrate The Guardian's triumphs during the Red War. It featured remade Redux Story Missions, three new armor sets per class to collect and upgrade, and new Eververse cosmetics. It also featured Destiny 2 Year 1 Moments of Triumphs.
List of appearances