If you ask an English speaker to spell a word, there are specific, widely-known names for all the letters to fill this need. The same appears to be true for all Phoenician-derived alphabets that I can think of, incl. Arabic, Hebrew, etc. (Although I don’t actually know whether they commonly use those names when spelling words since those names are longer than English’s — perhaps someone can enlighten me here?)
I think we can disregard syllabaries because the names for the letters here are obvious. Similarly for other scripts in which each lexeme constitutes a syllable (such as Han). Nevertheless, feel free to include interesting tidbits about this if you have any.
But what about other alphabetic/abjad scripts that are unrelated to Phoenician? What about the Brahmi scripts (Devanagari etc.)? Unfortunately I don’t know about many other scripts, but according to Unicode there must be loads. :) Can letter names for spelling be regarded as universal, or do some cultures really have no means of spelling out a word?