I too was looking for an answer to this and thought I'd add my findings to this question so I can find them again later (and hopefully they're of use to someone else too)
The URL of the SOAP API is $(MANTIS_URI)/api/soap/mantisconnect.php
(e.g. if your Mantis is usually accessed at http://localhost/mantis/
then go to http://localhost/mantis/api/soap/mantisconnect.php
)
It uses NuSOAP which means that if you point your web browser at that URL you get HTML documentation of the web service; most of the actions seem to have sensible names and a one-line description of what it does. The WSDL can be downloaded at $(MANTIS_URI)/api/soap/mantisconnect.php?wsdl
and, because I'm using C#, running wsdl.exe
on that URL gives me a basic strongly-typed client library for the service that seems pretty easy to work with.