Festival & Event Mounts
Festival & event mounts are only obtainable during special in-game events that Blizzard has scheduled throughout the year. As the timing of each event can vary from year-to-year (or month-to-month, in the case of the Darkmoon Faire and Timewalking) you'll have to check your in-game calendar for the precise dates. Some of these mounts are rare drops from special festival bosses - you can queue for these daily using the dungeon finder, during the event period. Others may be purchased with special currency that can be earned by participating in festival activities.
Brewfest occurs in late September and early October. Both Brewfest mounts are rare drops from a loot bag you receive after defeating the special festival boss, Coren Direbrew. You can receive either mount on either faction. You can queue for Coren using the dungeon finder during the festival.
The Darkmoon Faire opens for one week of each month. Check your in-game calendar for the next date. It can be reached from portals outside Thunder Bluff (for Horde) and Goldshire (for Alliance). Most of these mounts are purchased with a special currency - Darkmoon Prize Tickets - that are earned by participating in the various carnival games. The Skate can be purchased with Darkmoon Daggermaws which are fished up in the waters around Darkmoon Island. Check out Wowhead's Darkmoon Faire Guide for more information.
Timewalking events occur monthly, so check your in-game calendar for dates. During this week-long event you can queue for special Timewalking versions of dungeons from older expansions, with the expansion theme rotating each month. In Timewalking dungeons your level and gear are scaled down to the levels of the NPCs, and defeated bosses award a special currency - Timewarped Badges - which are used to purchase mounts and other special items. A bonus 500 Timewarped Badges may be earned each month by handing in a quest item which drops at the end of your first dungeon. The Timewalking vendors are only present during the event period, and the mount for sale changes depending on which expansion theme is active that month.
Feast of Winter Veil
Winter Veil occurs around Christmas-time - from mid-December to the start of January. You have a rare chance of obtaining the Minion of Grumpus from Savage Gifts that you can buy from your Draenor Garrison Events Coordinator during this period.
Hallow's End occurs around Halloween - from mid-late October until the start of November. The Headless Horseman's Mount is a rare drop from a loot bag you receive after defeating the special festival boss, Headless Horseman. You can queue for him using the dungeon finder during the festival.
Love is in the Air
Love is in the Air occurs close to Valentine's Day - around the second half of February. The two mounts available during this festival are obtained in different ways - the Swift Lovebird may simply be purchased with Love Tokens, which are earned by performing daily quests during the event, while the highly-coveted Big Love Rocket is an ultra-rare drop from a loot bag you receive after defeating the special festival boss, Apothecary Hummel and his coterie. You can queue for them using the dungeon finder during the festival.
Noblegarden occurs around Easter - roughly March to April. The purchase of this white tallstrider requires a quirky currency: Noblegarden Chocolates, which are gathered by looting and opening Brightly Colored Eggs during Noblegarden. The Springstrider may also be looted directly from an egg, but this is an extremely rare occurrence.