Admit that you've forgotten the trailing slash after umbraco a
hundred times.
Admit that it has made you use a thousand words your mother
never taught you.
Admit that you've had hundreds of calls from customers in the
same predicament.
Well... I have.
And all this time I've been convinced that there was no easy fix
- if there was, someone would have implemented it.
Today, I got fed up and just tried the simplest fix I could
think of - just for the heck of it.
Guess what? It worked! One XML tag with some attributes in the
rewrites section of /config/UrlRewriting.config and hey, presto,
the umbraco backend now follows the HTTP specification on how
trailing slashes should be handled - transparently.
Here it is:

EDIT: I'm very sorry that I have to give you the XML as an image
instead of using our newly installed syntax highlighter, but
ASP.NET barfs when trying to display the app-relative path (with ~)
in the destinationUrl when I try to do that, so I have no
other option (there is probably a fix... maybe I'll try to find it
some day)... :(
EDIT: Actually, what a HTTP-compliant web server does when it
encounters a "directory URL" that doesn't end with / is a redirect
to the same URL with the / appended. This causes an extra
round-trip but is by design. So, what my fix does is exactly what
the standard stipulates.