You can now register your bike online in the City of Santa Cruz (and get mailed the sticker) for free. This greatly reduces the hassle of bike registration, and is well worth the couple of minutes it takes.
You do need to know your bike’s serial number. The form is a bit weird, because they repurposed their incident report form software for the purpose:
I highly recommend that Santa Cruz residents register their bikes now, and that residents of other jurisdictions push their police departments, city councils, or whoever has the power locally to register bikes to also set up online bike registration.