Mounts in World of Warcraft: Legion Patch 7.1.5
(Including the opening of The Nighthold)
The savage Brawler's Burly Basilisk may be purchased at Rank 8 in the renovated Brawler's Guild. Sadly the old Brawler's Burly Mushan Beast is no longer available from vendors and it appears that it may have been retired unexpectedly.
The final two gorgeous falcosaur mounts are attainable now that The Nighthold raid is open. Both are awarded at the end of a long quest chain offered by their equivalent companion pets and both require the defeat of raid bosses from The Nighthold to complete. For more information on capturing the companion pets and starting the quest chains, see the detailed blog post by Quintessence of WarcraftPets.
The stunning green heavenly cloud serpent (last-remaining unused colour of this mount) is finally available! Yu'lei, Daughter of Jade may be purchased for 5000 Timewarped Badges, but the vendor is only present when the Mists of Pandaria Timewalking Event is active.
The Arcanist's Manasaber is available via a quest from Khadgar, Fate of the Nightborne, after defeating Gul'dan in The Nighthold - but only if you have already completed the Insurrection quest chain and picked up the extra quest, The Nighthold: Lord of the Shadow Council from Khadgar.
Two infernal mounts drop from the final boss of The Nighthold, Gul'dan, on different difficulties. The red Hellfire Infernal is a 100% drop chance on Mythic difficulty, and the fel-coloured Felblaze Infernal is a rare drop in regular and heroic modes.
The fel-green Steelbound Devourer is crafted by Blacksmithing. The plans may drop from Tichondrius in The Nighthold in at least regular mode and above, but aren't a guaranteed drop.