iPod5 to 9.3.1, now home button isn't working?

- Reset all settings

Go to Settings > General > Reset and tap Reset All Settings.

All your preferences and settings are reset. Information (such as contacts and calendars) and media (such as songs and videos) aren’t affected.

- Restore from backup. See:

Restore your device from an iCloud or iTunes backup - Apple Support

https://support.apple.com/en-us/HT204184

- Restore to factory settings/new iOS device.

If still problem, make an appointment at the Genius Bar of an Apple store since it appears you have a hardware problem.

Apple Retail Store - Genius Bar

https://www.apple.com/retail/geniusbar/

Unfortunately this is a fairly common issue with iPod touch 5th gen. Don't fret, this is not caused by updating to 9.3.1, and your iPod is not trash. You can use assistive touch as a on-screen home button, found under acessibility in settings, midway down the list.