Fairly comprehensive answer was provided in multiple other threads for databases which I direct you to them. In general, you can look at places like the crystallographic open database (COD) or American Mineralogist. SREC also provides a complete and upkept list of free crystallographic databases.
It may sound a bit arrogant...: Databases working in association with search-match routines should be only a rapid help to get ideas about the phases present in a specimen analysed by powder-XRD. Such database based methods should always should be combined with a good understanding of the chemical system under consideration, with a good understanding of the diffraction process (instrumental and how an overall pattern looks like, including effects of microstructure e.g. texture). Rietveld like refinement procedures are often an adequate method to confirm or disprove suggestions by the search-match routines. I have seen a lot of very meaningless outcomes of analyses of powder-XRD data by database based search-match routines...
The most respected and complete database is the Powder Diffraction File (PDF) from ICDD (International Center for Diffraction Data). But this is certainly not an open access database. It used to be called JCPDS (Joint Committee on Powder Diffraction Standards). The vast majority of good X-ray powder diffractometers are equipped with soft for phase analysis using this database.
Its disadvantage is that it is based on data that is correct from the point of view of an XRD specialist with a "light regard" to the quality of the samples from a chemical point of view.
I don't know "free" versions of JCPDS. In addition, there are at least 500,000 records (about 10 kB each - few terabytes total), a huge amount, you can’t write it on any DVD disks, so in recent decades only access to the database has been sale.