Odin has always been the one people wonder the most, and also troll the most. Lightning does not mean that Odin has spawned.

People claim there'll be blue lights in the sky, personally I've never looked so I'm not sure... But, the weather icon turns to a dark black/grey mist, and your entire screen will become greyscale (and for anyone in the Gridania maps).

The FATE he spawns is "Steel Reign: Lv 50", it does NOT show up on the map, but it DOES show on the Mini map. Meaning you need to be within' proximity to see the FATE on the mini map.
When he first spawns you also need to find him first, as he wont show up as a FATE - He starts off more like an NPC FATE where you need to interact with it first.

The areas and maps he spawns are random, there is no definitive cycle. Fallgouard Float, Bentbranch Meadows and Quarrymill are the three maps he'll spawn at though.