I'm at the point where if it requires a username and password I don't need it. Unless it's required for work or something I just don't have the mental space. I've had the same I-pod phone for almost 5 years and have zero apps

Ha, my brother is going to get me a free iphone 4, I am fixin' to move into the 21st century

Write them down and tape the list to the inside of my safe

I have a boatload of passwords...the only real way of keeping them secure is the old skool way--keep it in your head. Occasionally, I do forget a password for a site or system that rarely gets used, and in that case, I just use the forgotten password tool.

The NSA hacking your passwords is the least of your worries. Personally, I would NEVER put a password manager of any sort on the web--if something's on the web, people CAN get to it.

I use a combo of what has already been said:
Three passwords.
1. Super secure, in my case any banking/financial accounts.
2. Medium secure, Not too important but stuff I don't want anyone messing with, insurance, etc.
3. Low security, Pandora, personal email, etc.

Of course I still have a dozen or so logons so I need a document to keep track of what password goes where, that is in a nondescript word file.

But you don't do this for your daily systems- you will have a process of memory in place already. Anything else, it works.

Any new thoughts or methods in the past few years?

I have a system to remember most of them. Here is a more basic method.

Take a favorite phrase that you love, say TheQuickBrownFox. Perform some replacements. Say 0 for o, 1 for i, and # for e. You can go on and on with the replacements, but it gets complicated. It could be just a single favorite word, doesn't have to be a phrase, but pick a long and multi-syllable word.

Th#Qu1ckBr0wnF0x

Add some numbers before, after, or in the middle, that is significant to you. Say a spouse's birthday.

0101Th#Qu1ckBr0wnF0x1970

Then add something specific to the app or site. Say, TurboTax or WellsFargo.

0101Th#Qu1ckBr0wnF0x1970Turb0
W#lls0101Th#Qu1ckBr0wnF0x1970

Special Circumstances:

Some websites don't allow special characters like $, &, or %. Use capital letters to replace, say:

ThEQuick

Some websites have a maximum of 8 character passwords. Hopefully these aren't financial or government institutions, or anywhere that stores your SSN or credit card info... Just use a short word and some numbers.

If this is hard to remember, write your methodology, not the passwords themselves, in a password-protected file.

Post-it note on computer screen? :)

Still use the 'My Eyes Only' app. Neatly organized, always with me.

It is essential to my business.

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It's supposed to go under the keyboard, duh! ;)

My work password has to contain at least 8 digits, a mix of large and small caps, numbers, and at least two symbols and I have to change it every six weeks.

But my ATM code that protects my life savings is 4 numbers long and I haven't changed it since Clinton was president.

(ripped off from some comedian somewhere)

KeepPass is open source (you can check the code and compile it yourself if you're really paranoid).
I don't trust anything on the password services on the web .. cause its a matter of time before they're hacked and you don't know what the administrators are doing with the code..

I prefer Lastpass to the "post-it note" option for one major reason...the post-it note doesn't help me with my phone apps, or when I'm away on vacation and need to check my bank account, or at a friends house, or whatever. Lastpass will give me the password info at all of those places and more (and, it automatically fills in the fields, so i won't mistake a 1 for a l for a I, or a o for a O for a 0 in my notes.

a small paper notebook

Finally, a voice of reason...

DashLane. The point specifically is to have a different, random password for every different login, so if one is compromised, only one is compromised. Then TouchID unlocks it on my iOS devices. Then turn on two-factor for important stuff.

I use dashlane for my non-financial accounts, but it's a pain. I can't get it to work with gmail, so I have to enter those passwords by hand. I have a encrypted folder that's shared on all my devices that has screen shots of the login info for all my financial institutions.
I finally learned to record a new password BEFORE you commit to it. Too many times have I replaced a password and forgotten to record it somewhere.

I've been using Dashlane for about a year now. One recommendation - don't forget the password for it! There is NO recovery. After using it for about a week and manually entering about 50 or more login/pwd combinations for various apps/sites - I couldn't remember the password that I told myself I'd never forget. I had to download the app again and reenter all of the info again. I've stored the Dashlane pwd in a secret place on my phone content so I won't loose it again.