Within the US you can only be "listed" on 1 exchange. That listing exchange is considered the "primary" exchange.

Trades can occur anywhere: on-exchange, off-exchange, ATSs, over-the-counter OTC, etc...

But it gets more complicated when you go global as you can have cross-listings, as is common with Canadian stocks in the US. POT (Potash) for example is listed on TSX and cross-listed on NYSE... and can also trade on any other US exchange...