• Make sure the beastie you are looking for you have fully researched, it's not really possible for them to spawn rare without complete research.
  
• Have jerseys to spare, as long as you have no jersey that would be able to recruit that beastie it spawns as usual but it will never be shiny - they can be higher grade jerseys if need be but it is a waste of money, have variety for each level threshold just in case.

Do mind the process is heavily luck-based, this guide aims to increase the odds of finding one -doesn't mean they will be guaranteed though.
Remember that the first Raremorph missed per research number will be safely kept to be found later by an NPC named Indie, missing some in the early game or way later does not mean it is all doomed, it will only mean more hassle going to said NPC and then to the marked area trying to get it again. Note that this does not apply to duplicate Raremorphs, once it is lost after you already have one it is gone forever. (or until you find another one to recruit)

• Attract sprays - can be bought in any shop except SP ones - they will double the chances of finding rares as well as prevent your own reserves from showing up in areas you enter, usually spawning rarer % beasties. Note that the the level of the spray doesn't affect the odds increase, it only affects it's duration.
  
• When looking for a specific beastie it is best to go to an area where others appear less especially when avoiding repeated shinies, small maps will have 3 beasties.
  
• Bigger maps cause longer load times but also result in more beasties spawning in general, good place for increasing odds, more desired when looking for any rares rather than specific ones. This includes sailing as the entire area when on a boat is considered one big map meaning it's one of the easiest ones to get chance for any rare.
  
• Getting enough beasties researched (most likely the base 100 not including extinct ones) will result in the player getting a call and being given an Old Guidebook item which increases the odds - it is not clear how much does it increase them by but any increase to those chances can save time.
  
• Reese - once you have completed his friendship line you will be able to request a favor in form of a radar that causes chosen beastie to spawn more frequently and with higher odds of a rare. It does require you to put that beastie on your team for a moment when requesting it, you can switch it out once it's effects are active however. Odds increase is unknown, it is double useful for shiny hunting specific ones because it will both raise the odds for a rare, raise the spawn rate of selected beastie and reduce the odds of other beasties taking that place as the selected one will have it's % spawn increased. (It does not function as an actual radar, you will still have to find the beasties in rooms yourself as it only shows up as an icon in upper part of the screen showing that it is active)

• Smaller maps usually load in faster, resulting in more searches over a period of time, more desired when looking for specific rares than bulks of them.
  
• Higher spawn rate of a specific beastie in a room means it is more likely to be rare when visited repeatedly so some areas are better due to the naturally higher % chance of spawning, check the beastiepedia for a habitat of the beastie you are looking for, zoom in for % values to show.
  
• There is no limit to how many rares can spawn in a room, neither is there a timer bound to finding one and being unable to find more, the more chances to spawn there are the higher the possibility of finding them.
• The fastest way to reroll beasties is by leaving and re-entering between 2 rooms with desired spawns, cycling between one undesired back to the desired one makes the process twice as long especially if the room has no spawns (like Thousand Views). It is possible to reroll without leaving - spots that have camp sites or map teleports to them, with the second being faster as it skips the rest cutscene. Fastest way to achieve this is by standing right next to the marker or campsite, opening up the map directly with the map keybind and pressing confirm multiple times and as soon as it's possible you will re-enter the same room without having to run to the exit, it will spawn new beasties. Entering a boat does count as leaving - giving the chance for new beasties to be rare.
  
• To not be interrupted by away teams you can toggle them off in Settings menu in Online Features tab as having to fight them can be useful for money, but it does take time and they could get in the way of Raremorph searching.

Most important tell are the sound cues and your beasties reacting with a shiny icon if there is one currently in the area, the shiny sound can be heard instantly before even the game finished loading in the region fully if shiny is close enough to the entrance, and the sound is similiar to cartoonish sparkling effect, your beasties will also react almost instantly upon entering the room with a shiny icon with brief moments of repeating itself to make sure you don't miss out no matter the distance, as long as rare is present in the room no matter how big the area is - the icon is a sure way tell that one is there, it won't be in room next to it, it won't be in another area entirely, it is right now right here, as long as you don't cause the area to unload by leaving or by similiar actions.
As long as your beasties recognized there is a rare, there is one for sure, this includes beasties potentially hiding around like Webbounce, and they aren't triggered by your own rare reserves or team. This can have a side effect of being baited into fully checking a room when another rare is in the room and it is one you already have and/or don't want for some reason. Raremorphs will always be visible as the main team leader, they won't spawn as 3rd backup beastie for a normal morph team.

• Do not leave the area at all unless you are certain that beastie isn't worth the effort of getting it right now, this includes accidently walking out of the room because the cycle between room was so fast you couldn't react to the icon or the sound in time, give yourself some extra time to react just in case while still keeping your own pace. Leaving includes resting at a camp site, same room rerolls and entering or hopping off a boat! So bring healing items just in case as you will not be able to heal beasties by resting. Oddly enough boathouse and railhouse rests seem inconsistent with this rule.
  
• Do not approach it without the recruit condition plan covered and with auto battle mode on, it is easy to forget when grinding levels that it was left on, tap menu, settings, check if it's off, make sure your beasties have moves needed to get it to join, hit save in case things go wrong and then go for it.
  
• Do not win without the condition being met, losing, forfeiting or stalling the game as long as it's necessary until they are convinced to join is fair game, winning without fulfilling it will result in them despawning, where losing will not. Same turn recruit conditions are fine, for example dealing high damage with a serve, that serve can wipe the beastie you are trying to recruit as it will still count towards the recruit condition before the end of the game.
  

There are ways most likely covered in other guides, so i will let them guide you, but the special challenge points you get from expeditions and tower challenges, once you have way too many can give you a boost - you can simply buy athletic sprays, then sell them and buy jerseys along with the attract sprays you are actually in need of. The very basic one has the best points to money return ratio.
In case where audio doesn't cut it, go to areas with more visible entrance, most have multiple you can walk in from, picking the most visible one for your beasties is advised even if the time loss from loading and unloading could make small differences, you could despawn the beastie by not paying attention to it at all resulting in a trip to Indie.
You can adjust your team to have roughly the same or smaller in general size to easier see it, tall beasties can block the view of smaller ones and it seems to be decided randomly which one will speak up about the rare in the room which can lead to the tall beastie partially or entirely blocking the icon. Even removing most of your team to reserves if the 2 left will recruit for sure can help with visibility of the icon (mostly suggested for endgame with your team on very high level).
Similiar to how reducing the amount and size of beasties on your team it can also benefit your hearing of their specific noises, when the icon appears they will also make a noise based on the beastie the icon appeared on, getting used to your starters and others along the way can help you to recognize that sound easier, especially in area where those beasties do not normally occur.
Reducing the music volume or turning it down entirely while keeping the sounds doesn't just help with hearing the shiny noise which falls off with distance, but it also helps hear your own beasties more clearly as their sounds will be basically guaranteed to stick out as they are sure to appear right next to you on room entry. (Unless more quiet ones are part of the team)
Since they are extinct you will have to catch their other non-extinct morphs and evolve them into those, they do not appear as the ones you need to research for it and even if they were rare it would be impossible to catch them.

The advice above comes from getting all Raremorphs in the game, what the NPCs could tell you, and after that i went to verify some of the information - a YT video did cover half if not more of the things mentioned here by Indie, and parts of what i could find around the web seem to confirm what i experienced too. It was fun for the most part, grind got painful only at times - especially during the last 2 rares to find, it is strange to ignore most of the game to commit aggressive walking between rooms like it's a slot machine, but to anyone attempting it - goodluck, i didn't spoil it to myself by looking up how the rare ones would look and instead would be genuinely surprised after long time searching, some were really cool, some kinda worse than default, some i didn't expect in the slightest to look like that (in a good way). Keep it to moderate amount as to not risk burning out and it might be a fun alternative thing to do once you completed the game. Because i have seen so many Sefrens by this point someone shiny hunting will most likely have reserves that look like this too by the end of it: