When should I use a trailing slash in my URL?

It is not a question of preference. /base and /base/ have different semantics. In many cases, the difference is unimportant. But it is important when there are relative URLs.

• child relative to /base/ is /base/child.
• child relative to /base is (perhaps surprisingly) /child.

In my personal opinion trailing slashes are misused.

Basically the URL format came from the same UNIX format of files and folders, later on, on DOS systems, and finally, adapted for the web.

A typical URL for this book on a Unix-like operating system would be a file path such as file:///home/username/RomeoAndJuliet.pdf, identifying the electronic book saved in a file on a local hard disk.

Source: Wikipedia: Uniform Resource Identifier

Another good source to read: Wikipedia: URI Scheme

According to RFC 1738, which defined URLs in 1994, when resources contain references to other resources, they can use relative links to define the location of the second resource as if to say, "in the same place as this one except with the following relative path". It went on to say that such relative URLs are dependent on the original URL containing a hierarchical structure against which the relative link is based, and that the ftp, http, and file URL schemes are examples of some that can be considered hierarchical, with the components of the hierarchy being separated by "/".

Source: Wikipedia Uniform Resource Locator (URL)

Also:

That is the question we hear often. Onward to the answers! Historically, it’s common for URLs with a trailing slash to indicate a directory, and those without a trailing slash to denote a file:

http://example.com/foo/ (with trailing slash, conventionally a directory)

http://example.com/foo (without trailing slash, conventionally a file)

Source: Google WebMaster Central Blog - To slash or not to slash

Finally:

1. A slash at the end of the URL makes the address look "pretty".

2. A URL without a slash at the end and without an extension looks somewhat "weird".

3. You will never name your CSS file (for example) http://www.sample.com/stylesheet/ would you?

BUT I'm being a proponent of web best practices regardless of the environment. It can be wonky and unclear, just as you said about the URL with no ext.