"Incoming" relates to something that you are receiving or that is entering a location. "Oncoming" is usually to do with stuff moving opposite relative to the subject and has fewer uses. e.g. "Incoming traffic", traffic that is entering a building or park, "oncoming traffic", traffic that is moving in the opposite direction to you/the speaker And, "incoming e-mails", e-mails that you/the speaker receives, but no-one says "oncoming e-mails"...

I think there are a few usages not mentioned above: "The incoming Freshman class" (refers to the students that will be entering college next year) "Incoming mail" (mail that is going to be delivered to a specific location in the future) "The incoming mayor" (the mayor that will be replacing the old one during the change of office. Also known as the mayor-elect)

Oncoming is usually used in the context of traffic, and it is coming towards you. I was blinded by oncoming traffic, I swerved into oncoming traffic. Incoming can refer to missiles, so if one was heading for you someone might shout 'Incoming!' to warn everyone to take cover.