These days I am working quite a lot with two of Thomas Moore's important and influential song collections: both the Irish Melodies and the Popular National Airs. It is always helpful to have these publications available online and thankfully most of the original editions as well as some later complete editions have been digitized.
The Irish Melodies
There is no need to say here anything about this one, it is surely one of the most popular and most successful song collections ever published and a lot of these songs are still well known:
- A Selection of Irish Melodies. With Symphonies and Accompaniments by Sir John Stevenson MusDoc and Characteristic words by Thomas Moore Esq., 10 Volumes, J. Power, London, 1808-1834
The first two volumes are available at the Internet Archive. They belong to the Drs Whitby Music Collection by the University of Western Ontario, one of the best collections of historical sheet music and songbooks I have ever seen in the Internet:
The Bayerische Staatsbibliothek in München has also already digitized a considerable part of its massive musical holdings. Recently they have added the first 7 volumes of the Irish Melodies to their digital collection. They also have a copy of Vol. 8, but for some reason that one hasn't been scanned yet. I hope they will add it in the near future:
- 2 Mus.pr. 532-(1-7) [Click on Einzelbände to see the individual volumes] (also available at the Internet Archive)
Volume 8, 9 and 10 of the original editions are to my knowledge at the moment not available online. Therefore it is necessary to use one of the complete editions of Moore's collection. The Internet Archive has for example - among others - these three and they are all useful:
- Moore's Irish Melodies. With Symphonies and Accompaniments by Sir John Stevenson and Characteristic Words, Oliver Ditson & Co., Boston, n. d. 
- J. W. Glover (ed.), Irish Melodies with Symphonies and Accompaniments by Sir John Stevenson Mus. Doc. and Characteristic Words by Thomas Moore Esq., New Edition, Dublin, n. d. 
- Moore's Irish Melodies. With the Celebrated and Unsurpassed Symphonies and Accompaniments of Sir John Stevenson, Mus. Doc. and Sir Henry Bishop. Illustrated by Twenty Original Steel Engravings, After W. P. Frith, R. A., A. Elmore, R. A. &c. With a Biography of Thomas Moore and An Essay on the Music of Ireland, The London Printing And Publishing Company & A. W. Cittens, London & New York, n. d., 2 Vols
The Popular National Airs
The great success of the Irish Melodies encouraged Moore and his publisher to try out the same formula for national airs from all kind of different countries and in 1818 the first volume of this collection appeared. Here he offered new lyrics with melodies that were described as Indian, Spanish, Portuguese, Sicilian, Venetian, Scotch, Italian and Hungarian. Sir John Stevenson was again responsible for the music. Five more volumes would follow and from No. 2 onwards Henry Rowley Bishop wrote the arrangements:
- A Selection of Popular National Airs with Symphonies and Accompaniments by Sir John Stevenson MusDoc; Henry R. Bishop]. The Words by Thomas Moore, Esq., 6 Volumes, J. Power, London, 1818-1828
Again the BStB, München helps out with scans of the first three volumes:
- 4 Mus.pr. 35243-(1-3) [click on Einzelbände; also available at Google Books: Vol. 1, Vol. 2, Vol. 3 ]; also at the Internet Archive
Unfortunately they don't have copies of Vols. 4-6 and as far as I know no other library has yet digitized them. Therefore it is necessary to use a complete edition of this collection. I found only one that is available online:
- Charles W. Glover (ed.), National Airs, with Words by Thomas Moore, Longman, Green, Longman & Roberts, London 1860
This is the so-called "People's Edition of Moore's National Airs" and the editor has simplified the arrangements a little bit. In the preface he notes that "it has been my study to arrange the symphonies and accompaniments in the simplest appropriate form, so as to render the whole easy of execution". Nonetheless this edition is very helpful and I have used it quite a lot.
By the way, I am somewhat surprised that the National Airs have rarely been discussed by the Moore scholars. There is no critical study and as far as I know nobody has yet tried to identify the tunes and its sources. But this was an influential and groundbreaking and also very successful collection. Some of the songs became really popular. Even the reviews at that time were very positive (see f. ex. The Quarterly Musical Magazine And Review, 1, 1818, pp. 225-229 & 5, 1823, pp. 67-74, The Gentleman's Magazine 90 I, 1820, p. 521):
"This is certainly one of the most pleasing collections of the kind we ever recollect to have met with. We have, however, less to do with the music itself, than the delightful poetry which accompanies it, and which comprizes, according to our ideas of beauty, some of the most highly polished specimens of the art of Songwriting we know in the English language [...]".
A rival publisher apparently liked it so much that he hired songwriter Thomas A. Bayly as well as Bishop and Stevenson for a competing collection with the title Melodies of Various Nations (4 Vols, c. 1822-30). But Moore's work was also known outside of Britain and served as a model for other editors interested in these kind of international national airs. Friedrich Silcher from Tübingen, one of the most important German producers of "Volkslieder", used more than 20 tunes from Moore's Popular National Airs (as well as some more from the Irish Melodies) for his own Ausländische Volksmelodien (4 Vols., 1835-41, at the Internet Archive). Some of these German versions - with translations or new words mostly by poet Hermann Kurz - became very popular, for example "Stumm schläft der Sänger" (H. 1, No. 1; i. e. "Here Sleeps The Bard"), a song that even today still belongs to the repertoire of male choirs.
The Irish Melodies and the Popular National Airs have of course also been included in numerous collections of Moore's - more or less - complete works. But in most cases the music has been left out. But there is one massive edition where the tunes were included:
- The National Moore. Centenary Edition Including the Airs of the Irish Melodies, National Airs &c And a Memoir by J. F. Waller, William Mackenzie, London & Dublin, n. d.  (pdf available in the online catalog of the British Library, [select "I want this"])
These are 700 pages of Thomas Moore's works - I can't say if it is complete - and for every song not only from the two collections discussed here but also from others the tune has been included. Not at least this is a very beautiful book and very enjoyable to read and leaf through and well worth the download.