If you can dynamically justify a paragraph on the screen, the most common use of the hyphen is to divide words that do not fit on a justified line of type. Usual texts are treated different than poetry, anyhow, we use the hyphen ‘-’ to separate words (not sentences) and/or to create compound words, write numbers, adding certain prefixes, etc.

James and Alessandro silently let pass by, the term is not "hyphenation" for words. Hyphenation is when a word itself breaks.

“the black cat is still around,
playing with the blue ball”

or

“the black cat is still around, playing
with the blue ball”

I come across this problem a lot (in typesetting, that is) with the large font size I use for chapter titles. I prefer personally to set the top line longer than any subsequent lines, hold "the"'s together with the next word, and ... well ... whatever looks best. (Sometimes it takes several tries).

For a web page, I don't think I could be bothered :-)

You can tack words together with & nbsp; -- the non-breakable space (for example, "the blue"), or use the HTML < nobr > .. < /nobr > tags, which work just like InDesign's "No Break" feature. But you have to consider that with all different browsers, font sizes, and window widths, virtually everyone will get to see a different broken paragraph. Imagine doing this for every possible bad break ...

Besides, as current browsers do not use any intelligence at all, breaking paragraphs comes down to "Oi, the line is full. Go to next line." SO, if you use one of these tricks, the text will look worse instead of better.