Kinsey Millhone, from Sue Grafton's Alphabet books, is an obvious candidate.
How about Dorothy Sayers' Harriet Vane? She's listed as a protagonist in the wikipedia pages for the later Wimsey books...
A couple more series from Seanan McGuire:
Another older entry, recently remembered: Liath, in Kate Elliot's Crown of Stars series.
The Dr Who 13 announcement prompted me to record a few more additions to this list:
Of the assorted Forgotten Realms books I read during my school years, two of the main ones I remember liking were Spellfire and Azure Bonds, both featuring female leads (although I don't remember much in the way of details about either book - the D&D stuff tended towards "enjoyable" rather than "memorable").
I've been wandering through a variety of urban fantasy series of late, resulting in a couple more names to be added to this list:
Emily Rodda's "Pigs Might Fly" is another one I recall liking a lot as a primary school kid (I probably read it when I was around 8 or 9)
A couple more childhood ones that aren't on the bookshelves any more: Nancy Drew & Trixie Belden.
George was also definitely my favourite of the Famous Five, and while Lotta wasn't the main character in the Mr Galliano's Circus books, she was the most interesting. (As an adult, I'm aware there were lots of other problems with Enid Blyton's writing, but that all went completely over my head as a kid)
Those are the titles that occur to me off the top of my head. I'll likely come back and extend this list in the future after actually taking a look at my book shelves - the titles above are the ones where I didn't even need to look in order to remember them.
A series on my list to get back to is Kylie Chan's trilogy-of-trilogies featuring Emma Donahoe. Unlike the US based urban and post-apocalyptic SF&F mentioned above, this series is also notable for being primarily Hong Kong based (with an Australian element due to Emma's presence), and for having its supernatural concepts based in Chinese mythology rather than European.
Chan is fond of cliffhanger endings to books (and trilogies!), but doesn't quite always nail the "still resolve enough threads to be satisfying" aspect of that approach, so I'm currently waiting for the final trilogy to be fully completed before getting back to them.
Vin, the protagonist of Brandon Sanderson's Mistborn series is another favourite, although I confess that the thing I actually love the most about the Mistborn series is the elegant design of the magic systems (whilst many fantasy authors are happy to invent just one magic system, Sanderson invented three for Mistborn, weaving them together masterfully. The magic system in his first novel, Elantris, is different again).
Jim C. Hines's take on Cinderella (aka Danielle), Snow White (aka, well, Snow) and Sleeping Beauty (aka Talia) in the Stepsister Scheme and subsequent books is well worth a read (and you'll never look at the older versions of those stories the same way again).
I like Georgia Mason, from Mira Grant's Newsflesh series, as an example of a character who handles situations more through wits and foresight, than through anything else.
Verity Price, from Seanan McGuire's InCryptid series, on the other hand, is more likely to just hit things (but only if they had it coming).
Kate Daniels, from the eponymous series by Ilona Andrews, is brilliant. Also some fascinating world building here, in relation to the way the Shift impacts the structure of post-apocalyptic Atlanta and elsewhere.
That series isn't finished yet though, so if you do decide to pick them up, you'll eventually also end up stuck waiting for the next one, just like me!
Kat Richardson's Harper Blaine, in the Greywalker series, is a more recent favourite.
(The series is complete now, and well worth picking up if you think you might be interested in a bit of a detective-noir take on modern urban fantasy)
Heris Serano and Esmay Suiza anchor Elizabeth Moon's Serano Legacy series, while Ky Vatta anchors her Vatta's War series.
Sassinak, the lead character in a book Moon co-wrote with McCaffrey is another favourite, and in fact the stepping stone from McCaffrey's work to Moon's in my reading history.
Sorka was my favourite viewpoint character in Anne McCaffrey's Dragonsdawn, and Lessa and Menolly were firm favourites in the larger Dragonriders of Pern universe.
Eskarina Smith is the earliest female lead character I can recall that I'd still identify as a favourite protagonist. I read Terry Pratchett's Equal Rites when I was somewhere around ten or so, and she's still one of my favourite characters ever. It long annoyed me that she hadn't reappeared in any later Discworld novels, so it delighted me when she finally did (even if it was mostly a cameo appearance in someone else's story).
Esk's absence was also made much easier to tolerate through the presence of the wonderful Granny Weatherwax, Susan Sto-Helit and Tiffany Aching throughout various parts of the series.
Really, Discworld should just be required reading for everyone. It's proof that something that can be both tremendously educational and tremendously entertaining, all at the same time.
I sometimes wonder just how much reading Equal Rites when I did has to do with my intense dislike for helping to perpetuate broken systems. I certainly use Granny's "headology" lessons on an almost daily basis.
The idea of caring about whether the protagonist of a novel is male or not is one that's fairly alien to me, but I've gotten the impression that a lot of folks prefer to read novels where the main character shares the gender they themselves identify with.
This stream is about putting together a list of some of my own favourite female protagonists from books and series I've read over the years.