UIUC's own Hymn Tune Index is a database that "contains all hymn tunes printed anywhere in the world with English-language texts up to 1820, and their publication history up to that date." The HTI is entering its fourth decade as a preeminent resource for hymnological research. It began in the 1970s as the work of Professor Nicholas Temperley, who was conducting research on a book concerning English parish church music. By the early 1980s, HTI's database management structure was already in place, and subsequent decades saw several major updates and expansions, the last occurring in 2006. In 1998, Oxford University Press published the Print Index in four volumes.
HTI offers options to search by Tune, Text, Source, or Composer. Searching by Tune allows you to find music by the HTI Index number, the tune name, or the incipit. To find tunes associated with a particular text, use Text Search and words or phrases that start, are contained in, or exactly match portions of text. One can use Source Search to find items by HTI Source Code; Compiler; or Title; and Composer Search using either the composer's last name or his or her attribution in a source. As you can see below, the HTI search fields come with helpful examples to boot (just in case you're forgetting how to format your tune name or designate an incipit).
An excellent summary of the HTI's selection criteria and explanation of the scope of the database can be found here.
The Cyber Hymnal
The Cyber Hymnal contains the text to over 13,000 different hymn tunes searchable by various musical criteria. Some of the works also contain links to audio files as well as musical scores. Finally, the Cyber Hymnal also has links to brief biographies of the composers.