I have some trust issues though with offline wallet generators, say a paper wallet generator, or the vanitygen-master that generates a public key starting with for example 1Loyce. Without totally understanding the source of the program, how can I ever be sure it uses a random search for a new address? Untrusted software could generate predefined keys, so a third party can access it too. And for me as a user it is virtually impossible to know for sure what software I can trust.
Do you have trust issues with online wallets such as Electrum, MultiBit, Bitcoin Core, and Armory? Without totally understanding the source of the program, how can you ever be sure it uses a random number for a new private key? The wallet could generate predefined keys, so a third party can access it too, and for you as a user it is virtually impossible to know for sure what software you can trust.
Or, you could just use well reviewed open source software so that you can feel confident that any malicious programming would be identified and pointed out by other programmers. If you trust open source wallets, then you can trust open source offline wallet generators, paper wallet generators, and vanity wallet generators for the exact same reasons.