Please see my comment to the question.
The only reasonable answer I can give is: singleton pattern may or may not be used. This pattern should not be used often. In most cases of the application, it should not be used at all. So, what's the role of it in a Web application? No role at all, if you don't use this pattern. If you use this pattern, that is the role you gave to it. :-)
No special recommendations, except perhaps one:
In an ASP.NET application,
try not to add any singletons in addition to those which already present in ASP.NET technology (or application environment), such as
HttpContext
. Please see this useful discussion:
http://stackoverflow.com/questions/2134511/asp-net-singleton[
^].
See also:
http://msdn.microsoft.com/en-us/library/system.web.httpcontext%28v=vs.110%29.aspx[
^].
—SA