In Harry Potter and The Philosophers Stone, Harry, Ron and Hermione have serious trouble finding some information about Nicolas Flamel in the Library:
They had indeed been searching books for Flamel's name ever since Hagrid had let it slip, because how else were they going to find out what Snape was trying to steal? The trouble was, it was very hard to know where to begin, not knowing what Flamel might have done to get himself into a book. He wasn't in Great Wizards of the Twentieth Century, or Notable Magical Names of Our Time; he was missing, too, from Important Modern Magical Discoveries, and A Study of Recent Developments in Wizardry. And then, of course, there was the sheer size of the library; tens of thousands of books; thousands of shelves; hundreds of narrow rows.
Looks like they were just picking books and hoping to find something in them.
However, libraries have various techniques to solve this problem. For example library catalogs (Library Catalogs) or classification systems (Library Classification). These systems have existed since the 18th century.
The only "sort of" catalog in there is Madam Pince but in general she cares more about keeping silence than helping the students.
So why are there no index cards or catalogs in Hogwarts Library?
Since there was a lots of discussion on the topic - here a short summary:
- Looks like I overestimated the usefulness of library indexes - several people shared their own experience - if you don't have much information to start with, it is still a long search.
- JKR make Madam Pince's character unpleasant and unhelpful for plot purposes. This of course can not be considered a "plot hole" since unpleasant and unhelpful librarians do exist in the real world too (I have met some :) ).
- It is plausible to assume that actually Hogwarts library had more or less a decent index but due to the limited information that was available at the start and the unhelpful librarian, the Trio had trouble finding anything.