Syllabic slurs are those slurs that you see in vocal music, joining all the notes that are sung to a single syllable.
I don’t actually like them, myself. The world would be better off without them, for two reasons. One, they don’t tell you anything you don’t already know. A singer knows how long the syllable lasts by referring to the underlay, the hyphens between syllables and the lyric extension lines after final syllables. These things show unambiguously when you change from one syllable to the next, and you can’t avoid seeing them because you have to look at the words. So the slur joining the notes doesn’t add to your knowledge.
And two, given that slurs don’t tell you anything you don’t already know, they are just clutter, and especially on a busy page, you don’t want clutter. A page without syllabic slurs looks fresher than a page with loads of them.
Not only that, but if you are using slurs to echo the underlay information, you don’t use them for the purpose God intended, which is to show phrasing. A quick glance at vocal repertoire will show that most vocal music doesn’t show phrases with slurs. Is that because slurs have come to have this other, less important, role in vocal music? Or is it a sign that actually there’s no conflict at all between the different uses of slurs, because everyone uses them to show underlay, and no one to show phrasing? After all, is not the phrasing obvious from the words themselves?
So now I’ve accidentally argued myself into a situation where slurs aren’t required for phrasing in vocal staves, and aren’t required for underlay, because all of that information comes from the words. So, like a magician whose trick has gone disappointingly wrong at the last minute, I’ve just made slurs on vocal staves utterly redundant.
Hmm, that’s not what I meant to do. From now on I’ll just do what Elaine Gould says, which is this: Use slurs for phrasing if the composer has put them in (though they almost never do), otherwise use syllabic slurs.