Most but not all trumpets have serial numbers. The larger and more well-known the manufacturer, and the more recent the manufacture date, the more likely it is that the trumpet has a serial number.
Many manufacturers began consistently marking serial numbers on their trumpets in the last 50 years.
Conn-Selmer brands typically include serial numbers on their trumpets. The Conn-Selmer brands include Armstrong, Artley, Bach, Benge, Conn, King, and Selmer. Olds also include serial numbers on their trumpets, starting in 1928.
However, there is the chance that the trumpet is from a less well-known manufacturer who did not stamp serial numbers, that the number has worn off, there was no stamp given because of a manufacturer error, or that there was some other reason the trumpet did not get a serial number.