How to Unlock Shadowlands Flying in Patch 9.1
When you start Patch 9.1 at Renown 40, you will begin a new series of Covenant campaign chapters -- 9 in all. Good new: completing all of them isn't required to fly!
Renown 44 opens Chapter 12, The Last Sigil, the third Covenant Campaign chapter in this second series of campaigns. At the end of this storyline, you will get the quest  The Primus Returns and receive the reward  Memories of Sunless Skies.
Clicking  Memories of Sunless Skies will teach you and all characters on your account Shadowlands Flying!
Where You Can and Cannot Fly in Shadowlands
Shadowlands Flying will allow you to fly in the following zones:
Bastion
Maldraxxus
Ardenweald
Revendreth
Shadowlands Flying does not allow you to fly in the following zones:
Oribos
The Maw
Korthia
The In-Between
You will not be able to fly from one continent to another; you will still need to use flight points to get from one Covenant zone to another.
Covenant Flying Mounts in Patch 9.1
9.1 also includes new Covenant-specific Flying Mounts, which become available at Renown 45. There is also a Renown 70 mount, and a Covenant-only mount purchased from Threadbare Cloth in Korthia for 1000 and Exalted reputation with Death's Advance, and a Covenant-only mount that drops off a specific Rare mount in Korthia.
If you wish to collect the flying mounts for all four Covenants, you will need to complete this requirement for each Covenant, but these mounts will be accountwide for any alts who belong to the same Covenant.
The Kyrian mounts:
 Elysian Aquilon - a flying Aquilon Centurion mount earned at Renown 45
 Ascendant's Aquilon - a flying Aquilon Centurion mount earned at Renown 70
 Battle-Hardened Aquilon - a flying Aquilon Centurion mount from Death's Advance rep
 Foresworn Aquilon - a flying Aquilon Centurion mount from  Intact Aquilon Core, acquired from rare Korthian Aberration Wild Worldcracker
The Necrolord mounts:
 Maldraxxian Corpsefly Harness - a flying Corpsefly mount earned at Renown 45
 Regal Corpsefly Harness - a flying Corpsefly mount earned at Renown 70
 Battlefield Swarmer Harness - a flying Corpsefly mount from Death's Advance rep
 Lord of the Corpseflies - a flying Corpsefly mount from  Legend of the Animaswell, acquired from rare Korthian Corpsefly Fleshwing
The Night Fae mounts:
 Ardenweald Wilderling Harness - a flying Wildering mount earned at Renown 45
 Autumnal Wilderling Harness - a flying Wildering mount earned at Renown 70
 Winter Wilderling Harness - a flying Wildering mount from Death's Advance rep
 Summer Wilderling Harness - a flying Wildering mount from  Wilderling Saddle, acquired from rare Korthian Wilderling Escaped Wilderling
The Venthyr mounts:
 Sinfall Gravewing - a flying Gravewing mount earned at Renown 45
 Obsidian Gravewing - a flying Gravewing mount earned at Renown 70
 Pale Gravewing - a flying Gravewing mount from Death's Advance rep
 Mastercraft Gravewing - a flying Gravewing mount from  Gravewing Crystal, acquired from the No Stoneborn Left Behind event
Ground Mounts Will be Usable in the Maw in 9.1
Starting with Patch 9.1, all ground mounts will now be usable in The Maw after completing a quest. Flying mounts will still be grounded until you open Shadowlands Flying. Also, this effect is not account wide, so every character who wishes to use ground mounts in the Maw will need to complete this quest.
Shortly after you enter Korthia for the first time, you will encounter the attendants responsible for the City of Secrets. In the quest  Who is the Maw Walker? (part of the story line Maw Walkers), your anima will be analyzed. This will awaken the power of the True Maw Walker within you, allowing you to summon any of your mounts in the Maw -- and by extension, in Korthia.
As a bonus, the True Maw Walker effect opens Perdition Hold and Beastwarrens in the Maw without needing to grind rep with Ve'nari.