Oddities are unlike standard Foundables in Harry Potter Wizards Unite, in that they are not freed from Confoundables the usual way.
Their Traces spawn on the on the Map, just like you would see other Traces from the Exploration tab of the Registry — except their appearance is tied to certain conditions that must be met.
This is where Wizards Unite's time of day and weather mechanics come into play.
Here is the current list of Oddities available in the register, and the conditions you need in order to find them:
Please note that Dawn, Day, Dusk and Night refer to your local time, so it is best to check when the game moves from one phase to another in your time zone in order to more effectively chase those Oddities!
(Source)
Several other Oddities are included in the game code, such as Death Eaters, Three-Headed Dogs, Prisoners of Azkaban, and more. They will likely be added at a more advanced stage of play.
These creatures will be represented on the map by purple Traces. The beam of light above the Trace represents the Rarity Level.
If you notice an error in the spawn rates listed above, don't hesitate to let us know in the comments!