What filesystems does iOS use?
The other answer seems to be correct about the file system being HFSX*, but on being case-insensitive (other answer since edited to correct), I found that not to be the case in the book Mac OS X and iOS Internals. On page 23 it says:

In iOS, being the case sensitive HFSX by default, case is not only
preserved, but allows for multiple files to have the same name, albeit
with a different case. Naturally, case sensitivity means typos produce
a totally different command or file reference, often a wrong one.
[emphasis above is mine]

Since that didn't agree with the other answer, I sought confirmation.  I loaded a sample iOS project I was working on and changed the name of the database to only differ by case.  After running the app on my iPad mini (iOS 6.1) to re-create the database, I went into iTunes and looked at the files under File Sharing for the application.  Here's what I saw:

So, it would appear Mac OS X and iOS Internals is correct: case is preserved and allows for multiple files to have the same name.
Interestingly, when I selected both files in iTunes' File Sharing and clicked &quot;Save to...&quot; my desktop, I did not get any warning.  iTunes apparently &quot;saved&quot; both files to my desktop, but only the initial-uppercase version survived (presumably overwriting the initial-lowercase version of the file.)

* 2017 Update to Apple File System (APFS):
As of iOS 10.3, released on March 27, 2017, iOS now uses the new Apple File System (APFS), said to be &quot;optimized for flash and solid-state drive storage, with a primary focus on encryption.&quot;
APFS was announced at WWDC 2016 and there is a WWDC video titled Introducing Apple File System, and an Apple File System Guide for developers. The features section of the guide may be of specific interest.