the advantages of not using the .aspx extension:
- hides your technology
- makes technology changes easier
- shortens the url
- not really any SEO benefit other than possible improvement in url "clickability"

a trailing slash implies a "directory-type structure".
if you have a url such as http://example.com/widgets/ then you should also have content at urls such as http://example.com/widgets/red and http://example.com/widgets/blue.
http://example.com/widgets/ should be the "directory index" for /widgets/ on your site.
otherwise, you should forgo the trailing slash - http://example.com/widgets.
not really any SEO benefit here either - more of an information architecture issue.

you should never use the default directory index document (such as default.aspx or index.html) in the url - it's the default.

the most important SEO issue is using the canonical url for internal links and redirecting all non-canonical requests to the canonical url.
this applies to using the file extension vs. not, using the default index directory document vs. not, and using the trailing slash vs. not.
in each of the 3 cases only one of the two choices is canonical.

Thanks, phranque.

>>> if you have a url such as http://example.com/widgets/ then you should also have content at urls such as http://example.com/widgets/red and http://example.com/widgets/blue.

I thought I need to have: http://example.com/widgets/red/ and http://example.com/widgets/blue/ The way WP uses it. Or I'm missing something?

Extensionless means without .aspx and without trailing slash.

The trailing slash indates a folder, or the index page within a folder.

Don't use a trailing slash for the URL for a page.

But a trailing slash is being used on WordPress sites, say mashable. I.e. each page ends with "/".

and joomla will prevent you from creating that trailing slash url should you wish to do so.
just because the CMS is incapable doesn't make it correct.