Please note the tipping structure defined below.  This question may be
answered for from $2.00 to $10.00 depending on information provided.

There exist many software packages that can be used for issue
tracking, such as GNATS, ClearQuest, Razor/PT, etc.  Most of these
solutions involve installing a server on your company intranet, and
allowing users to access the server through a client or web browser.

My desire is to find a company that provides a turnkey solution where
the server is hosted on a computer in their domain, and we can allow
users both within our company and from our partners to access the
data via a web client.  The server must provide SSL encryption or equivalent.

A cursory examination has uncovered http://www.bug-track.com/ as one
possible solution that will fit this need.  I am looking for a list of
similar services, preferably one that is focused on the "leaders" in
this market (if there is such a distinction).

A satisfactory answer to this question will be a list of five
providers of this solution.  A list of ten or more providers, with
some sort of measure of leadership (number of users, marketshare,
etc.) will earn an additional $3.00 tip.

A highly desired feature will be a service that provides, in addition
to some sort of web client, an API through which queries can be
submitted and the underlying data can be changed via scripts.  An
answer that identifies three or more solutions that provide such an
API will earn an additional $5.00 tip.