There is no specific lane for assassins, it depends on what type they are and how you play them. Often champions can be played in multiple ways, so the definition of assassin is quite broad anyways, some bruisers like Irelia can be played as assassins or as bruisers.
Usually such burst champions are played in mid or top lane because they need the experience from a solo lane since that is where a majority of their burst comes from. However, some assassins don't need the experience that much but rely more on the element of suprise so they are played in the jungle instead.
Top: Poppy, Pantheon, Xin
Mid: LeBlanc, Evelyn, Fizz
Jungle: Nocturne, Shaco
Top or Mid: Talon, Akali
Most mids can be played just as well at top if they got counterpicked at mid.