A Summoner can find the right chest solo, with two other people along for the ride. Keep Carbuncle out at all times. If a chest turns out to be a mimic, Carby will begin attacking it while the SMN runs away. Once the SMN is far enough away (40' is safe) they can Release and resummon, then continue to the next chest. The old mimic will deaggro.
Keep in mind that the Mimic will not deaggro back into a chest immediately. If you resummon Carbuncle as soon as you release him, there is a chance he will run back to auto-attack the mimic. If this transpires, just use your "Retreat" ability and Carbuncle should return and all will be fine.
A player with Summoner subjob can do this trick too, though not as well; Garuda with Aerial Armor can buy you enough time to get away.
Occasionally the Mimic may get in a few lucky hits, killing Garuda too soon and causing the player to be drawn in. Because of this, it is recommended that any /SMN should be accompanied by the rest of the party to help kill the Mimic.
The Video to the right only demonstrates what to do, as a Summoner, if you encounter a Mimic. It is expected that everyone would know what to do if they came across the correct chest.
A Beastmaster can also solo this using a similar (yet more complicated) method as Summoner. Call out Courier Carrie (due to his high defense and low cost, she's perfect for this assault), and buff yourself with Protect/Stoneskin/Blink/Utsusemi if you have it. Find a Cursed Chest, and try to open it. If it turns out to be a mimic, get your pet to fight it. Now, run away quickly and place yourself so that you can see the red dot of the Mimic just outside the radar's outer circle. Use Heel from here, the Mimic's hate should transfer to you, but it won't be able to draw you in from that distance. Your pet will be able to escape with minimal damage and you can try another Cursed Chest.
The radar estimation was done using a non-Ninja radar. You will have to adjust your position accordingly if subbing Ninja. Placement is very important; you have to be within hate range so your action gains the Mimic's attention; but far enough out to be out of Draw In range.
This method could require some practice, its definately not as easy as SMN until you get the hang of it.
The mimics have a limited aggro range. You can Reraise without aggro if you make sure you do not die directly on top of a mimic. As long as you're about 4' away, you can Reraise safely after the mimic turns back into a green text Chest. Since the mimic only draws in every couple of seconds, you can make sure you die at a safe distance if you're careful to time your final run away from the mimic before you die.
Since the Pugil pursue by scent and do not link, it is possible for a job such as RDM or BLM to aggro a Pugil and escape. Sleep them (they are not resistant to sleep), then cast Deodorize and run away. They will not be able to track you if you put plenty of distance between yourself and it.
It Seems the Fish can not go into the cave that you start in so as a Summoner if you get aggro have your pet attack it and run to the cave and release your pet and it will follow you to the cave but will no go inside and will lose hate and walk back to its starting spot.