There are multiple companies that make trombones, a brass instrument characterized by a sliding piece on the front end (see picture). These companies -- Bach, Conn and Yamaha, just to name a few -- all have different designs for their instrument, and each style has its own corresponding model number. While the model number is not printed on the individual instrument, there are several methods of determining this information.
Local the manufacturer's logo or insignia on your trombone. Different manufacturers have different logos, and print these logos at various locations on the instrument; for example, the King Instruments company prints a crown on the outside base of the trombone's bell (the flared part of the instrument from where the sound comes). A Conn trombone prints the word "CONN" on the round part of the third brace (the short, straight metal rod nearest the back of the instrument). Once you know who manufactured your trombone, you will be able to narrow down your search for the instrument's model number.
Take note of the different features on your trombone. Is it composed of brass (which will have a golden color) or sterling silver? Measure the diameter of the bell, as this is a key measurement often used to distinguish trombones. Look to see if the trombone is operated by a slide -- as most traditional trombones are -- or by valves, like you'd find on a trumpet (Reference 1).
Visit the manufacturer's website and look for its "trombone" section. Some companies will list their different trombone models online, while others will have a downloadable PDF catalogue. This is the information for which you are looking.
Compare your trombone's features with those listed for the manufacturer's various models.
Use the trombone's serial number to help identify the model number. Regardless of your trombone's manufacturer, the serial number should be located on the piece of metal where the slide attaches to the rest of the instrument (Reference 2). Knowing your trombone's serial number can help you identify the year your instrument was made. King and Conn have lists of serial numbers and corresponding years of manufacture; this information can help you narrow down your search even further.
Contact your local music store with the information you've learned about your trombone. Even if you don't have enough information about the instrument for you to find the model number online or in the manufacturer's catalog, the addition of a serial number should help an instrument expert find it for you.
You may not be able to locate the manufacturer's logo or insignia on the instrument, especially if the parts where this information was printed have been replaced. In this case, you will have to rely on the trombone's features and serial number for model number identification.