When is it appropriate to use non-breaking spaces?

The usage of a non-breaking space is explained in a Wikipedia article under Non-breaking spaces and Controlling line breaks and below in items 1 and 5:

1. It is advisable to use a non-breaking space (also known as a hard space) to prevent the end-of-line displacement of elements that would be awkward at the beginning of a new line:

• in expressions in which figures and abbreviations (or symbols) are separated by a space (e.g. 17 kg, AD 565, 2:50 pm);
• between the date number and month name (e.g. 3 June or June 3);
• in other places where breaking across lines might be disruptive to the reader, especially in infoboxes, such as £11 billion, June 2011, 5° 24′ 21.12″ N, Boeing 747, after the number in a numbered address (e.g. 123 Fake Street) and before Roman numerals at the end of phrases (e.g. World War II and Pope Benedict XVI).

2. A hard space can be produced with the HTML code   instead of the space bar; 19 kg yields a non-breaking 19 kg.
3. A literal hard space, such as one of the Unicode non-breaking space characters, should not be used since some web browsers will not load them properly when editing.
4. Unlike normal spaces, multiple hard spaces are not compressed by browsers into a single space.
5. A non-breaking space should be used before a spaced en dash.

One place where it's important is between a person's initials (e.g., J. Q. Adams), so that the J. and Q. don't end up separated. (A nonbreaking space isn't needed between the initials and the last name, though).