Have there been any attempts made to make a comprehensive list of all scrolls or pieces of scrolls of the New Testament (in original languages) that have ever been discovered by archeologists?
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For the New Testament, yes. Though Wikipedia may not be a "scholarly" source I find it good for this sort of thing.
For the Greek Old Testament (i.e., "Septuagint"), there is also the Verzeichnis der griechischen Handschriften des Alten Testaments [= Catalogue of the Greek Manuscripts of the Old Testament], originally compiled by Alfred Rahlfs and published in 1914. It is available from Archive.org, or in a hi-res PDF from the Göttingen Septuaginta-Unternehmen.
It was revised in 2004 by Detlef Fraenkel -- see the SBL bookreview for a description of the work, and the changes from Rahlfs' original edition.
[I know OP is interested in New Testament manuscripts, but this catalogue has many manuscripts that also contain NT material, and comments on the question suggest a slightly wider interest beyond the limits of the question as posed.]