Recent changes to Spanish legislation have created a situation where a simple paper document (Certificado de autorización de residencia) is replacing the traditional residency card (Tarjeta de residencia). The new certificate is no good for identification purposes, so you may be needing your passport more often in the future. Holders of old-style residency cards can keep using them as ID documents until they expire.
Please note that in order to register in the local census (padrón), you need to show the following:
A mains bill that certifies the ownership of the property (water, electricity, IBI/SUMA bill, telephone bill etc.) or your Title Deed (a Copia simple carrying a blue notarial stamp is normally accepted).