How do you divide the syllables of plurals whose singular ends in "-e"?

If you are trying to decide whether you should break a line at applianc-es or applian-ces, don't use either one. They both look bad.

It's just two or three letters. You should generally be able to either fit them on the same line with the applianc- part, or break the word appli-ances. For confirmation, the TeX typesetting software, which usually knows what it's doing when it comes to hyphenation, gives ap-pli-ances as the possible hyphenation points.

By analogy with the -er and -ing endings, the appropriate hyphenation would be applianc-es. Merriam-Webster hyphenates fencing as fenc-ing and silencer as si-lenc-er. But the suffix -es is only two letters, so it usually shouldn't be necessary to break before it. And if we believe the TeX typesetting software, it generally isn't done.