[Edit]

I don't know about the Classical era but libraries would start to multiply in the late Middle Ages with the rise of universities.

A university like the Sorbonne had to maintain a solid library for its students and teachers. Foreign clerks were also welcome and books could be sometimes borrowed.

You'll find a lot of information on early libraries in the following books edited by André Vernet: "Histoire des bibliothèques françaises. Les bibliothèques médiévales du VIe siècle à 1530". Including for example the chapters written by François Dolbeau ("Les usagers des bibliothèques") and Marie Henriette Jullien de Pommerol ("Livres d'étudiants, bibliothèques de collèges et d'universités").

It is also true that the fashion for books mostly originated from France in the late Middle Ages and Early Modern Times...

As university educated clerks started to serve in the administration of kings and princes, the aristocratic elite started in France to amass books and found private collections (14th-16th centuries). Books were kept by the same officers who would be in charge of the safekeeping of jewels at the Burgundian court.

From then on even lords sometimes made their private libraries accessible to clerks at their service, or to friends and relatives.

With the invention of the printing press, university and aristocratic libraries grew exponentially. The rise of an administrative monarchy and type of government also made the contact of books and documents a primary necessity in many day-to-day matters.

Religious order and universities basically held the first public libraries and we still have many of their catalogues today. We sometimes know about their reading room rules and prescriptions. We also have accounting evidence to track down the value of books back then.

Claude de Grandrue for example wrote the Saint Victor Abbey (Paris) library catalogue and its books were stamped with the coat of arms of the religious order. This is the case of this copy of Ovide moralisé (an old french translation of Ovid's Metamorphosis): https://gallica.bnf.fr/ark:/12148/btv1b10500676w/f13.item.zoom You can see the abbey coat of arms on the top of the folio and if you go to the very first folio you'll see the book original shelfmark (KKK.29) with a latin description of its content.

Medieval library catalogues were edited very early by German and Austrian scholars (respectively: the "Mittelalterliche Bibliothekskataloge. Deutschlands und der Schweiz" and the "Mittelalterliche Bibliothekskataloge. Österreichs"). Belgian scholars have just finished to publish their Corpus Catalogorum Belgii in seven volumes and the Corpus of British Medieval Library Catalogue is already well known. Such massive works give us a lot of information on early private and public libraries.

[After a few hours of sleep and once I'm connected to my computer I'll be able to provide better references.]