I have only seen the Great Litany set to the usual tone as found in the L.U. / Graduale.
As for other Litanies I am sure that I have seen the Litany of Lorreto set to 5 different tones, Sacred Heart 2 or 3, and St. Joseph 2. All found in books published post 1900, so I would suspect that other tones of the Great Litany may be found.

Q. Where does one look for tones of the Great Litany? Perhaps looking for the various online Graduale / Notated Missale and looking at The Vigil and Of course the rogations Mass and procession (Greater and lesser Litanies).

I have had a quick look at the Sarum books and have not found the Great Litany, I will try again.

@tomjaw: What is the other tone for the litany of St. Joseph, and can you provide a score & scan? I have only ever seen the one that is in the LU, and many other editions.

I am mainly interested in case it fits within the same parameters as the litany tones which I am planning to include in my never finished book-project (basically, I'm sticking to stuff from Solesmes editions, omitting tones that are not quite in line with what seems to be the standard paradigm).

I was able to purchase both of these online for a reasonable price, many thanks for the lead.

That alternate tone for the Litany of St. Joseph I was able to also find online HERE. However, I'm a little puzzled by the endings of the Agnus Dei's, which do not follow the expected accentuation pattern.

The one I think I will be wanting to include in my book is a tone for the Litany of the Sacred Heart that I had been wondering about in another discussion HERE.

Richard's original question about on which litany tone Byrd based his setting now finds a decent answer in the Sarum Processional. There are several litanies (see the vigil of the Ascension for example), but Byrd here seems to set the Rogation litany, with a few accidental or stylistic differences.

Have a look through the whole document - there are many interesting chants, including a troped Salve Regina!