Recently I was researching some Danish and Norwegian tunes that I had found in German songbooks. Therefore I had to make myself familiar with the most important early collections of "folk-songs" from these two countries. Here is a little overview with links to the most useful digital resources. This is mostly about books including tunes, not collections of only texts.
Interestingly the earliest examples of Scandinavian "folk tunes" can be found in a French publication:
- Jean-Benjamin de Laborde, Essai Sur La Musique Ancienne Et Moderne, Tome Second. Livre Troisieme. Abrégé d'un Traité de Composition, Paris, 1780 (available at the Internet Archive)
This massive Essai was of course one of the most important musicological publications of the 18th century. One chapter in Book 4 is dedicated to the "Chansons du Danmark, de la Norvege & de l'Islande" (pp. 397-418). Of course Laborde hadn't made a field trip to Scandinavia. His informant was C. F. Jacobi, at that time secretary of the Kongelige Videnskabers Selskab in Copenhagen, who sent him an interesting collection of tunes and songs from these countries.
In case of Denmark I have to mention one very early important text collection: Anders Sørensen Vedel's Et hundrede udvalde Danske Viser, first published in 1591 and then updated and expanded by Peter Syv in 1695 (200 Viser og Konger, Kemper og Andre, a later reprint, 1739, is available at Google Books and the Internet Archive). Vedel's and Syv's work was a starting-point and source for all later editors.
In 1810 literature historian Knut Lyne Rahbek compiled a little book with the title Danske og Norske Historiske Mindesange (available at KBK). In the following years he then put together and published with two other scholars the first comprehensive modern collection of old Danish songs.
- Werner H. Abrahamson, Rasmus Nyerup & K. L. Rahbek, Udvalgte Danske Viser fra Middelalderen; efter A. S. Vedels og P. Syvs trykte Udgaver og efter handskrevne Samlingar undgivne paa ny, 5 Vols., Schultz, København, 1812-1814 (at BStB-DS: 10049001 P.o.rel. 4820-1(-5) )
Thankfully Vol. 5 included a selection of tunes (pp. XVII-LXXXVIII). This edition was also discussed outside of Denmark, for example in an interesting and detailed article about Alte Volksmelodien des Nordens in the Allgemeine Musikalische Zeitung (Vol. 18, 1816, pp. 594-599, pp. 613-619, also Beylage No. 6, No. 7). Collections of Danish songs were also published both in Germany and in England:
- Danish And Norwegian Melodies. Selected by A. Andersen Feldborg, of the University of Copenhagen, Harmonized and Arranged with Additional Symphonies and Accompaniments for the Piano-Forte, by C. Stokes. The Poetry Translated by William Sidney Walker of Trinity College, Cambridge, Chappell & Co., London, 1815 (at IMSLP)
- Friedrich L. A. Kunzen, Auswahl der vorzüglichsten altdänischen Volksmelodien und Heldenlieder mit Begleitung des Pianoforte, Wenzler, Kopenhagen, n. d.  (at IMSLP)
Nyerup and Rahbek were the key figures of the first "Folk revival". Later their work was of course eclipsed and overshadowed by Svend Grundtvig. The first volume of his great collection Danmarks gamle Folkeviser appeared in 1853. But he only published the texts but no tunes. More relevant for the musical side of this genre became A. P. Berggreen (1801-1880), a multi-talented composer, writer and editor. His first relevant publications were a songbook for schools and a collection of patriotic songs:
- A. P. Berggreen, Sange til Skolebrug, udsatte for tre Stemmer, 14 Vols., Reitzel, København 1834-1876 (at KBK)
- A. P. Berggreen, Melodier til de af "Selskabet for Selskabet for Trykkefrihedens rette Brug" udgivne fædrelandshistoriske Digte, C. C. Lose & Olsen, København, 1840 (at KBK)
But soon the first edition of his great collection of national and international "folke-sanger" appeared. A second expanded edition was published during the 1860s:
- A. P. Berggreen, Folke-Sanger og Melodier, Fædrelandske og Fremmede, Samlade og Udsatte for Pianoforte, 8 Vols., C. A. Reitzel, København, 1842-7 (parts of this 1st ed. are available at University of Michigan, Scores Collection); 2nd ed., 10 Vols., Th. Lind, København, 1860-1870 (at BStB-DS: 4 Mus.pr. 473-1(-11) & Sibley Music Library; at the Internet Archive: Vol. 1 & Vol. 2 )
In Norway the collection and publication of so-called "Folk songs" started much later than in Denmark. Of course there were Norwegian songs and tunes in Laborde's Essai, for example a couple of Edvard Storms Døleviser, long before they were published at home. Then one should not forget the legendary Abbé Vogler who also collected the tune of Storm's "Skogmøte af Torjer Skjeille" and then presented it to his German audiences as "eine alte Weise von den Gränzen von Grönland" (in his Polymelos, 1806, see Verzeichnis Falter, 1810, p. 23). Later this particular melody was used by Carl Maria von Weber and then - thanks to Friedrich Silcher - it became the tune of a popular "Volkslied". But this is another story that I am busy writing at the moment.
The above-mentioned British edition by Stokes had also included some Norwegian songs but it took some more time until the first real collection of Folkeviser from Norway appeared:
- Jørgen Moe, Samling af Sange, Folkeviser og Stev i Norske Almuedialekter, Malling, Christiania, 1840, at BStB-DS: P.o.rel. 5333 n & at Google Books)
This was basically a collection of texts, but young composer Ludvig M. Lindeman (1812-1887; see Norsk Biografisk Leksikon) compiled an appendix with some tunes. Lindeman became the most important collector and editor of Norwegian folk tunes. His first relevant own publication (with arrangements for male choirs) was:
- Ludvig M. Lindeman, Norske Folkeviser udsatte for fire Mandsstemmer, A. Th. Nissen, Christiania, n.d.  (pdf at Lindemanslegat.no)
- Ludvig M. Lindeman, Ældre og nyere Norske Fjeldmelodier. Samlade og bearbeidade for Pianoforte, Förste Bind, P. T. Malling, Christiania, n. d.  (at IMSLP & Internet Archive; the complete set has been re-published as a facsimilé in 1963)
More of Lindeman's works - some of them digitized - are listed on the site Lindemanslegat. Two of them should be added here:
- M. B. Landstad, Norske Folkeviser, Chr. Tönsberg, Christiania, 1853 (at Google Books; incl. an appendix with tunes compiled by Lindeman [pp. 869-920])
- Ludvig M. Lindeman, Halvhundrede Norske Fjeldmelodier harmoniserede for Mandsstemmer, Fabritius, Christiania, 1862 (pdf at Lindemanslegat.no; another collection with choral arrangements).
And of course I must mention again A. P. Berggreen. The second volume of his above-mentioned collection is dedicated to Norway. The first edition appeared in the 40s and the second in 1861:
- Jens Henrik Koudal, Rasmus Nyerups visearbejde og folkevisesamlingen 1809-21, in: Musik & Forskning 8, 1982, pp. 5-79 (online at http://dvm.nu/periodical/mf/mf_1982/)
- Olav Solberg, Editionen von Balladen und Volksliedern im Norden, in: Paula Henrikson & Christian Janss, Geschichte der Edition in Skandinavien, Berlin etc., 2013 (= Bausteine zur Geschichte der Edition 4), pp. 97-124