In SQL Server, this type of index is called a filtered index.1
Partial indexes have been supported in PostgreSQL since version 7.2, released in February 2002.2
SQLite supports partial indexes since version 3.8.0.3
MongoDB supports partial indexes since version 3.2.4
SQL Server 2008 Documentation: Filtered Index Design Guidelines. Microsoft TechNet. https://technet.microsoft.com/en-us/library/cc280372.aspx ↩
"PostgreSQL Documentation: Release Notes: Release 7.2". PostgreSQL. Retrieved 2009-10-09. Enable partial indexes (Martijn van Oosterhout) http://www.postgresql.org/docs/current/static/release-7-2.html ↩
"Partial Indexes". Retrieved 2014-02-04. http://www.sqlite.org/partialindex.html ↩
MongoDB V302 Release Notes. https://docs.mongodb.org/manual/release-notes/3.2/ ↩