How to get statistics about Bitcoin users? Because each Bitcoin user can operate multiple Bitcoin clients and wallets, and not every user runs the client constantly, it seems very difficult to gather data about the Bitcoin community. 

It's clear that many people in the community will want to remain anonymous (for good reason, if they hold a large amount of value in their wallet), and that collection of statistics in the Bitcoin client app would not be a popular option. Having read about malware that mines bitcoins, it's even possible that botnets may run a bitcoin client too. Even if such a collection technique was palatable, it couldn't be depended upon.

If statistics were gathered on a per-user basis rather than a per-client basis, a whole new data set could provide valuable information to the community, such as growth, feature popularity/desirability, usage preferences etc. Is there any mechanism with which a substantial quantity of real user data could be (or is being) collected reliably? Facebook comes to mind as a possibility, but even Facebook is easy to create fake accounts in.
The closest thing I'm aware of is RowIT's Bitcoin Stats Page which counts listening hosts, unique IPs and several other statistics which come closer to, but still does not directly equal, a count of users. It is, at least, less likely for one user to run a client from two IP addresses than to simply run two clients from the same address, so unique IP statistics might be somewhat more useful to you.

Realistically, Bitcoin was explicitly created with privacy in mind, so as much data mining as we are capable of, there are certain statistics you're simply unlikely to ever lay hands on.