OK, here's my list, although in doing further research, it seems only 2 or 3 of them are public domain, with a further few that come from operas -- you might be able to do with a vocal score + a clarinet orchestral part so you don't have to buy someone else's arrangement. Then I will list the ones that you definitely have to purchase.
PUBLIC DOMAIN
Schubert, Shepherd on the Rock (IMSLP has this, but look under the German, Der Hirt auf dem Felsen)
Spohr, 6 German Songs, op.103 -- IMSLP doesn't have this, but I would think they should be available somewhere
Meyerbeer, Das Hirtenlied -- the dates would make this PD, but I can't find it in any download site
FROM OPERAS: these will have originally been for the singer on stage, with obbligato clarinet and string accompaniment. They will not be public domain because whoever did the reduction from string accompaniment to keyboard accompaniment will have the copyright. However, there is nothing to stop you from using a vocal score to the aria, and just inserting the orchestral part.
Mozart -- 2 arias from Clemenza di Tito -- "Parto, parto" and "Non piu di fiori". These are both originally for basset clarinet, but it's easy enough to bring up the lowest notes with intelligently maneuvered transitions. Unfortunately, IMSLP doesn't have vocal scores or the clarinet parts yet. However, if your pianist is skilled enough to read an orchestral score, you can do a reduction that way. (I think Parto is Act I, Scene 9, and Non piu is Act II but I'm not sure where.
Rimsky-Korsakov, Song of the Shepherd Lell from the Snow Maiden -- IMSLP has the full score of this opera. I don't know what act this is in, partly because Lell has several important songs. It might be the 3rd song, for which IMSLP also has a vocal score, but this is transposed to a different key, so the clarinet part wouldn't match. You could still use the score to get a reasonable working version.
Spohr wrote an opera called Faust, and an aria in it is "Ich bin allein" for soprano, clarinet strings. This opera is very obscure now, IMSLP doesn't have it, so you will probably have to purchase the voice/clar/pno edition.
SACRED WORKS
Schubert, Totus in Corde Lanqueo -- from the First Offertorium, D.136. I can't find a PD download of this anywhere, but theoretically it could be available.
Cherubini, Ave Maria -- this was written in 1816, and Kalmus publishes it, though I think with English Horn substitution. The original was clarinet, and I got a PD download, but I can't find the web page now. It was from someone who did their own re-arrangement back to the original.
NON-PD WORKS (you will have to purchase)
Bononcini/Emperor Joseph I -- Viennese Arias. These are baroque, and good examples of our early history. Originally chalumeau.
Bliss, 2 Nursery Rhymes
Cooke, Three Songs of Innocence
Crusell, From Ganges' Beauteous Strands (might be from an opera, but to my knowledge, Crusell never wrote one . . . )
Herrmann, 3 Deutsche Lieder for soprano (or baritone), clarinet, piano
Jacob, 4 Seasonal Songs for voice, clar, piano
Jacob, 3 Songs for voice, clar (no piano in this one, but that might make an interesting texture change in a recital)
Kreutzer, Das Muhlrad
Vaughan Williams, 3 Vocalises for voice and clarinet (no piano, but again, might make an interesting change)
The rest of my list adds other instruments in.
Hope this helps. If anyone else has others to add, I'm always interested in expanding my lists!