Holding page please expand/edit/collate, Suggest a listing or Category of RE/ME's later as we get more information.
Random Encounters are a VIP only feature where an NPC character contacts the player and gives them an item or a task, which upon the completion of the task will grant them some goods.
Random Encounters and Miracle Encounters are often lumped together in game parlance. Technically Random Encounters are tied to a single scene and are shown by a Lantern Icon above the head of the NPC that grants them, being occasionally visible when the R.E.'s conditions have been met.
A "Miracle encounter" is where an NPC appropriates the player and tasks them with something in any scene.
Currently it appears that spending time in a given scene performing tasks and gifting NPC's with goods increases the likelihood of getting an RE but as they are by definition "Random" this cannot automatically cause one to occur for your character.
Generally the game music is muted when you are subject to an R.E. and replaced with a pulsing sound not dissimilar to a beating heart. You must act within about one minute to actually accept the R.E. or you may lose it.
Many of the games most sought after items are only available via R.E. Should you be lucky there will often be a world announcement that you have received such an item, so do not be surprised if random offers to buy it suddenly appear at that point!