I hate playing the "guess the password" game. Hitting "Forgot Your Password?" is even worse, because then it's "Reset" and one more new password to remember.
Is there any safe way to keep a record of all the passwords I currently use?
Is it safer to use a single password across the 10+ password-protected sites that I regularly use?
Lastpass is an excellent and secure password management tool : https://lastpass.com/
Keepass is another alternative if you prefer to keep them out of the cloud: http://keepass.info/
I would strongly advise against using the same password over multiple sites.
Come up with your own personal algorithm that is different for each site, but has a core phrase or word you can remember.
For example:
www.yahoo.com base phrase is: oatmeal
password would be: 5oY.comOatmeal
5 = letters in URL
o = last letter of URL
Y = first letter
.com = suffix of site
Oatmeal = your phrase that stays the same between sites
I have many passwords (no one account, aside from a few I don't really care about, has the same password and most of my passwords are 20 characters long (mix of upper, lower and numeric).
I use keepass to keep track of them all. There are several such password managers out there. Make sure you get one that encrypts its contents and make sure you don't forget or lose the access credentials for that.
I use Password Safe myself; there are plenty of similar tools out there.
Most of my passwords are randomly generated. I use, for example, 12 or more random lowercase letters when I can (easier to type on mobile devices), though some sites require a mix of upper and lower case letters, digits and punctuation.
I never use the same password on two or more sites. It's easy enough to generate and save a new password that there's no advantage in doing so.
I don't even remember the vast majority of my passwords, because I don't need to.
This does mean that I need access to my password database, and that database is potentially a single point of failure, but it's worth it. The database itself is, of course, encrypted (with a password that I have memorized); I keep it on a USB thumb drive and keep backup copies in safe places.
You didn't say what these passwords protect. What would be the consequences of one getting leaked? What you do for a low-value site might not be at all appropriate for a high-value one.
A password manager application like Keepass/KeepassX (the one I use), LastPass, or 1Password works well for a lot of people. With only a few, you might not need that.
Writing them down isn't terrible if you keep it in your wallet and don't identify the sites they refer to, AND IF they don't protect any valuable assets. Do this for your forum logins, for example, but not your bank or e-mail passwords.
Or make up a simple, memorable, way to create a password from a site name which isn't obviously reversible. Again, only do this for low-value sites.
