There is no complete directory of competitions. Even publications dedicated exclusively to cataloging competitions are not exhaustive. Many sources exist to research further, and some of them are listed below.
British & International Music Yearbook
A print volume is available at the above link.
The Countess of Munster Musical Trust: A Quick Guide to Funding & Competitions Available for Various Groups of Young Musicians
A British site listing competitions for soloists and chamber ensembles.
A Directory of Major International Classical Music Competitions
Indiana University Libraries
Links to about thirty competitions.
Making Music in a Looking Glass Land
Devotes a chapter to the advantages of participating in competitions and describes strategies for choosing events and participating successfully. Published by the Concert Artists Guild.
Musical America: International Directory of the Performing Arts
The web site contains industry news and resources. The print edition, available at the web site or an academic music library, contains numerous resources including a directory of competitions.
Organized by category, focuses on upcoming events.
MusicalOnline Competitions and Festivals
Lists competitions and festivals by state and country.
Open Directory Project
Sketchy for now.
Wikipedia list of Music Competitions
A list of over 100 competitions.
World Federation of International Music Competitions
This site lists contact information and laureates for all of the member organizations. You may search by category or by location. Most of the members are European.