This in no way resembles any sort of comprehensive list of all known Classical manuscripts. I wish such a list had been gathered conveniently between book covers, and I could just refer you to the title and ISBN.
Maybe such a list exists. I haven't found it yet. What I have found is some running totals of the numbers of manuscripts known for this or that author. I've found some of these figures in volumes I've had for a while. They've often been hiding in plain sight in the footnotes, where I only recently thought to look.
In A Companion to Homer, ed by Wace and Stubbings, London: MacMillan, 1962, on p 229, in the footnotes to JA Davison's chapter "The Transmission of the Text," we learn that TW Allen had listed 190 medieval and post-medieval manuscripts of the Iliad in his 1931 edition, including 7 which also include the Odyssey, that Allen had listed 75 manuscripts of the Odyssey in volume V of the Papers of the British School at Rome, including those 7 already mentioned, for a total of 258 medieval and post-medieval manuscripts of Homer. Davison' notes also mention ancient manuscripts of Homer listed in RA Pack, Greek and Latin Litrerary Texts from Graeco-Roman Egypt, published in 1952: 381 manuscripts of the Iliad and 111 for the Odyssey. That adds up to a nice round total of 750 manuscript of Homer. Davison points out that these figures do not include quotations of Homer in the works of other authors, nor indirect sources.
And remember, this was in 1962. I would imagine that more Homeric manuscripts have come to light since then. How many more? I dunno. Can I provide an example of even one specific discovery made since 1962? Strangely, I cannot. There's a ton of stuff online about Homeric manuscripts in general and Homeric papyri in particular, and from my personal point of view, none of it is user-friendly.
In Die Platonhandschriften Und Ihre Gegenseitigen Beziehungen by Martin Wohlrab, published in 1887 in Leipzig by Teubner, page 643, Wohlrab says that his survey includes 147 manuscripts. (This Teubner volume is a reprint from an academic journal, and begins on page 643.) Also on p 643 Wohlrab said that surely many more manuscripts of Plato would be found. This was before the Oxyrhynchus excavations began. How many papyrii of Plato have been found at Oxyrhynchus? And down the road at Fayum? I dunno. Lots, I would imagine. But Wohlrab was talking about manuscripts laying around in libraries which hadn't yet been catalogued. Was he right, in the 1880, when he predicted that many more manuscripts of Plato would be found in libraries? I dunno. I would guess he was right.
In Texts and Transmission, ed by LD Reynolds, Oxford, 1983, on page xxvii, Reynolds counts up some surviving manuscripts of Sallust: 2 from the 9th century, 4 from the 10th, 33 from the 11th, 58 from the 12th, 39 from the 13th, 46 from the 14th and 330 from the 15th, for a total of 482, and adds in a footnote: "The figures are incomplete, especially for the later period." In addition to these medieval manuscripts of Sallust, there are 4 ancient papyrii. 486, but "the figures are incomplete."
On p 36 of Texts and Transmission, Michael Winterbottom mentions 162 recent and unimportant manuscripts of Caesar. I was unable to find a figure which included both the unimportant and the important manuscripts.
On p 412 of Texts and Transmission, Michael Reeve informs us that we have over 650 manuscripts of Terence and adds, "Published estimates stop at 450. I owe the new figure to Claudia Villa."
On p 394, Reeve mentions "over 160 manuscripts" of Statius' Thebiad. Just of the Thebiad. The total number of manuscripts of Statius is more. How many more? I dunno.
I don't know how many manuscripts there are of Cicero. I don't want to know. I'm not a fan. (There are lots and lots.)
And one more time for Reeve: on p 107 of Studies in Latin Literature and its Tradition: in Honour of C O Brink, Cambridge, 1989, he counts up 154 of the 3rd decade of Livy. That's just for the 3rd decade (books 21-30). The total number of Livy manuscripts is somewhat more. How many more? I dunno.
On p vi of his 2004 Oxford edition of Ovid's Metamorphoses, RJ Tarrant informs us that we have over 400 manuscripts of that poem. How many manuscripts do we have of all of Ovid's works? I dunno. Very many, I would imagine.