There are different ways to be able to get a rare mount.
One of them is the classic way of opening caskets and chests which drop from magic wand, gift of heaven, magic mirror etc. as well as opening chests from the special event "Valuable Chests" and as one keeps opening them, he/she will receive bronze, silver and golden horseshoes, as the player starts to get more horseshoes, the chances of him/her receiving a rare prize will increase. At some point, the player will receive a rare prize, this rare prize might be one of the three (scorpolion, cerrador or dzerug) rare mounts.
The second way is to play magic mirror, when the Special Event "Mirror Magic" the players can open 3 mirrors by paying 75 silvers, if player manages to get enough points, he/she might get a rare mount as a prize (Please note that the rare mount has to be in one of the prize slots shown there as every time they play, the prizes change).
Third way is to join the lotteries made by the administration such as anniversary lotteries where players can buy tickets for the lottery with money, diamonds or event tokens so that they have bigger chance of receiving one of these rare mounts.
Fourth way is to join special forum events and being the #1 such as "2016 Soccer Euro Championship" where players had to guess the match results as well as completing puzzles and answering questions.
Fifth way is by bidding on the mount when it's sold in Closed Auction by the administration. As far as I can remember, the last auction was done last year and -Pinky Princess- has paid the highest amount, thus received the Cerrador Amulet.
However, the ones I have mentioned above are only for Dzerug, Scorpolion or Cerrador.
The other rare mounts are received by the following ways:
Ash Vargash and its upgrades: Received in special event "Faeo Under the Control of Darkness"
Ice Bear: Received in Seasonal Event "December - Winter Time in Faeo" but unlike Ash Vargash, this one is not bought from an NPC directly, it's bought in the Closed Auction by bidding on one of three amulet lots made by administration. However, after succesfully buying the amulet, players don't have to come to closed auction again to be able to buy the upgrades, they have to buy them from the NPC in Fir Tree in City Square.
Players can also get this mount via Referral system. If one's referral receives level 12, the referrer will receive an Ice Bear Amulet.
Fierce Pkhadd: It used to be received by referral system, however right now the only way to receive it is by using the Magic Lasso received via Collections section (see below).
The last way i know of is by gathering different statuettes (as well as 99 ludial links to be able to finish the set) which are shown in the picture by doing different stuff in game. For example: To be able to receive Lucky Cerrador Statuette, the player will have to open caskets, there are 10 times and 70 times more likelyhood caskets made for this statuette which is sold in Valuable Chests event. For Lucky Scorpolion Statuette, players will have to play Magic Mirror, For Lucky Dzerug Statuette, players will have to spend money on Luck Glow event, For Lucky Ice Bear Statuette, players will have to either make 50 golds spending in Trade Fair, or open Reward Caskets received by doing different tasks (see Activity section in Events tab in game). For Lucky Aranag Statuette, players will have to use Magic Wand and lastly, players will have to score more than 7 points in Gift of Heaven to be able to receive Lucky Corvus Statuette. After gathering all of the statuettes and 99 Ludial Chain Links, players will get the Magic Lasso, by using the lasso in a certain location, the player will receive a rare mount. The places and mounts are listed below:
Fortress of Captivity: Cerrador
Plateau of Silence: Scorpolion
Haunted Place: Restless Dzerug
Iceberg Wharf: Ice Bear
Settlement of Chernag/Klesva: Fierce Pkhadd