I use LastPass to store and use my passwords, so I do not have duplicate passwords even if I have to register four to five different accounts a day, and the passwords are long.

How safe are password manager services like LastPass? Don't they create a single point of failure? They are very attractive services for hackers. How can I trust the people behind these services and their security mechanisms?
I imagine that a third party (government, company, etc.) would be very easy to 'bribe' and get all of my passwords. 

Are there any other solutions that offer similar services with similar ease of use?

We should distinguish between offline password managers (like Password Safe) and online password managers (like LastPass).

Offline password managers carry relatively little risk. It is true that the saved passwords are a single point of failure. But then, your computer is a single point of failure too. The most likely cause of a breach is getting malware on your computer. Without a password manager, malware can quietly sit and capture all the passwords you use. With a password manager, it's slightly worse, because once the malware has captured the master password, it gets all your passwords. But then, who cares about the ones you never use? It is theoretically possible that the password manager could be trojaned, or have a back door - but this is true with any software. I feel comfortable trusting widely used password managers, like Password Safe.

Online password managers have the significant benefit that your passwords are available on anyone's computer, but they also carry somewhat more risk. Partly that the online database could be breached (whether by hacking, court order, malicious insider, etc.) Also because LastPass integrates with browsers, it has a larger attack surface, so there could be technical vulnerabilities (which are unlikely with a standalone app like Password Safe).

Now, for most people these risks are acceptable, and I would suggest that the approach of using a password manager like LastPass for most of your passwords is better than using the same password everywhere - which seems to be the main alternative. But I wouldn't store every password in there; make an effort to memorize your most important ones, like online banking.

I know someone who won't use Password Safe and instead has a physical notebook with his passwords in obfuscated form. This notebook is obviously much safer against malware... whether it's at greater risk of loss/theft is an interesting question.
