As with any hardware, you need to supply a "brand name" or manufacturer and what the "model number" is. Of course, a model number is not a serial number. Even within a brand, each model may have its own set of serial numbers. I suggest you do a google search for "brand serial numbers" replacing the word brand with your specific brand. But here is a short cut that may serve you well:
If you want to try another type of direct google search for your:
If you do not know the brand, then try the type of instrument and serial numbers. That may offer you a list of different instrument brands with links to serial numbers associated with them. Here is a result for a search for clarinet serial numbers.
For example: http://www.woodwind.org/clarinet/Equipment/HowOld/