While at work, my internet access to some sites is very limited. I cannot access DropBox or use USB drive. Can you actually look up the passwords on your phone using these apps or are the passwords encrypted? Also, do any of these apps generate random passwords?
Speaking just for Keepass (though the others are probably the same) the data is encrypted on your phone but when you enter your Master Password you can then see what your user names and passwords are for the individual sites you saved.
Without having it installed on your work computer you can then look up on your phone what you need to enter for a particular site. I would be concerned though since your work limits the usb access and blocks Dropbox, that they may also be monitoring the websites you are going to and could even be recording the keystrokes you enter (BTW this is legally allowed in the US for an employer to do though they should have an internet policy in place).
If there are sensitive sites you need to go to while at work, use your phone then or wait until you get home.
What password manager do you use or recommended that is safe to use
I recently developed an app called Password Shield, check it out. All passwords are stored local to your device and encrypted AES 128bit. Has some neat features I haven't seen in any other password manager apps. Let me know what you think.
I use mSecure and it's decent. It could import my SplashID database from my Palm (I was afraid to buy the high-priced Android SplashID back then because of the terrible reviews and because I would also have to buy an expensive desktop companion just to migrate my Palm SplashID stuff).
Also, it has a free but rudimentary companion Windows backup/restore program (mBackup) that does the backup/restore over wi-fi, or you can pay for a better PC version that also allows editing, and IIRC, form filling on the PC's browsers.