There's a lot of misinformation floating around about vocal ranges for certain roles in musicals, especially "Joseph," because there are so many versions. These are the ranges for each role as written in the licensed score (note that this show is sometimes toyed with as far as musical interpretation in different productions, but this is how it appears literally in the script). Classifications like "soprano," etc., are my own judgement of who would be most comfortable and stuff singing each role. Further analysis is included as I saw fit.
Soprano or Tenor (Low A to High A).
Much of the Narrator's part is very high - "Go Go Go Joseph" is a song that especially lingers all over sustained F's and G's. There are also a few parts that require lower notes (for example the openings of each act and a good deal of "Joseph's Coat" all require strength below middle C). (S)he stays in the upper-mid range for most of the show; the vast majority of the Narrator's songs require high E's and F's.
Tenor (Low B-flat to High G).
Joseph's setup is similar to the Narrator's in that most of his stuff lingers on the upper-mid range, calling for strong E's and F's. The G comes in as his final note in the finale. Note that the first rendition of "Any Dream Will Do" is a step lower than the one in the finale, requiring him to have a solid B-flat.
Tenor/Baritone (Low B-flat to High A-flat).
Mostly lower, hovering around low B and B-flat a lot. But watch for the high G-flat and A-flat that might sneak right up on you.
Baritone (Low B to High D-flat).
No surprises. This is right in the midrange that basically everyone has solidly.
Baritone (Low D to B).
Not a huge range here. Have fun with the line though ("Joseph, I'll see you rot in jail...") - interpretations of Potiphar's role can range from hilarious to f-ing scary.
Alto/Mezzo (Low D to A).
Her only line in the score is "Come and lie with me, love." Often, a non-singer is cast who speaks the line in character. Either way works well.
*Reuben [Those Canaan Days]:
Baritone/Tenor (Low C to High F).
Most productions will undoubtedly opt to have all the brothers sing the refrain ("Those Canaan Days we used to know..."), but in reality Reuben is assigned almost the entire song as a solo in the score.
Benjamin Calypso Soloist:
Tenor (F to High F).
The brothers all sing the refrain ("Oh no, not he," etc.,) and the soloist has the verses ("I hear the steeldrums sing their song," etc.). Some productions will add the "la-LAAAAA" star note, which is a top B-flat.
One More Angel Soloist:
The verse that is given to the soloist in the score has a range of F to high F, but it's written to be spoken in meter, with X-ed notes.
Tenor (Low C to High G).
The Butler is all over high G, and has the second-tenor part in the first act finale. Because the licensed score is based on the Peter Reeves recording, he's still referred to as the "Lively Lad" in the Pharaoh sequence ("Then this lively lad said..."), but most directors will be on top of it and know that it should be changed to the Butler.
Tenor (Low D to High G).
All over high G, like the Butler. The Baker has the lowest part in the first act finale (actually, the chorus isn't listed at the very end of the song; it seems to be a trio between the Narrator, Butler and Baker. Most productions, however, will probably use the chorus here because it's simply more effective that way.)
Baritone/Mezzo (Low E to High C).
Soprano (F to Top B-flat).
Baritone/Mezzo (Low C to F).
* "Those Canaan Days" is the only of the three "brother solos" assigned to one in particular, Reuben. The other two simply say "One Solo Brother." Note again that the "One More Angel" solo is only written as one verse (the brothers all sing most of it together), and that is indicated to be spoken in meter.
** The Camels, Queen Victoria and the Sphinx make an appearance in between "One More Angel" and "Potiphar," singing variations of the "Poor, poor Joseph, whatcha gonna do?" refrain as he travels to Egypt.