How to find the trials:
All trials must be manually located, and often don't update to show on the map. Use the progression counters for each region on the world map to narrow down where any uncompleted trials are, and then fly around that region using Revelio (Revelio's range for objective icons is massive while flying, making it very easy to locate any uncharted trials). Regions and their progression counters for collectables can be seen by zooming all the way out on the world map.
How to complete the trials:
There are 9 types of Merlin trials, all being pretty straightforward and simple, however some require specific spells to complete.
Upon activating the trial, 3 statues will crumble apart. To complete the trial, you must cast Reparo on all three statues. I recommend trying to locate the three statues before activating the trial, as Revelio often has issues highlighting the debris of the statues (making them difficult to locate).
Upon activating the trial, you will need to destroy all of the spheres using a basic cast or any damaging spells to complete the trial. There will be 9 spheres in total, often located in groups of 3 around the trial's origin.
Upon activating the trial, you will need complete the miniature parkour course made up of large smooth stone blocks. You must touch the surface of every stone block consecutively without touching the ground or any other surface. You can simply complete the course on your broom by tapping the surface of each block.
Cube Flippendo Trial:
Upon activating the trial, you must locate the 3 pedestals with cubes on top of them. To complete the trial, you must cast Flippendo on any side of the cube until each symbol on all sides of the cube align with the symbols on the pedestal. All 3 cubes must be aligned with their pedestals to complete the trial.
Upon activating the trial, you will need to locate and destroy 5 stones. Destroying the stones can be done by casting Confringo, Diffindo, Bombarda, or Incindio (I haven't confirmed if Expelliarmus works).
Upon activating the trial, you will need to locate 3 basins on top of pedestals. To complete the trial, you will need to cast Confringo on all 3 basins before they sink into the water beneath them (a basin will begin to sink once it is lit). Some pedestals will be shorter than others (indicating they will sink quicker), so I would recommend igniting them in order from tallest to shortest.
Upon activating the trial, you will need to locate the giant stone ball and its counterpart, the stone "hole". To complete the trial, you must get the ball into the hole by casting Accio, Depulso, or Wingardium Leviosa on it. For one trial, the hole will be in shallow water making it difficult to locate, and in another it will be located uphill making it difficult to complete. Every trial generally has an intended path for the ball to follow.
Lumos Moth Trial:
Upon activating the trial, 3 swarms of glowing moths will appear around the area of the trial, sometimes hidden out of sight or behind destructible objects. To complete the trial, you must guide the moths using Lumos to the 3 stone lanterns around the trial's origin. Guiding the moths can sometimes be finicky, as climbing up cliffs will deactivate Lumos. I recommend locating the 3 stone lanterns before gathering the moths, so you don't end up guiding the furthest moth swarm to the furthest lantern.
Orb Stacking Trial:
Upon activating the trial, you will need to locate 3 stone platforms and 3 groups of stone orbs. To complete the trial, you will need to cast Accio, Depulso, or Wingardium Leviosa on the orbs and guide them to one of the platforms. The orbs will stack themselves onto the platform and encase themselves in vines to indicate they've been completed. You must stack all 3 groups of orbs on all 3 platforms to complete the trial.