Read-only API that serves up custom selected parts of the OSM map data. (Source Code / Wiki)
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operational issues about the Postpass instance run on postpass.geofabrik.de
Powerful API that allows to query and retrieve changeset information. (Source Code)
SPARQL API for OSM data. (Wiki)
Free extracts of OSM features as GeoJSON, filtering on tags and bounding box. (Docs)