I prefer to use KeePass. It store the password file on your computer, with 256-bit encryption. You can have plugins that sync the file with your phone, so you always have it on the go, it sync's with a Google chrome (or other web browser) extension allowing you to login through a web browser. You can have plugins to make the password pronounceable or made up of certain words and characters. LastPass has more of a risk of being leaked because they are stored on an internet server, this has hardly any, as it has up to 3 ways to unlock the encrypted file, including a master password, key file, and Windows User Account. There is also a portable version of KeePass, which can be stored on a USB, and it does not require install on other computers.
The best part is, it's absolutely free!
Link to KeePass: click here
Verson 2.24 Professional edition is the best to get, not classic edition.