No mention of Blockchain.info?
They are one of the few online wallet providers that give users access to their own private keys. This means that even if they were to shut down, you would still be able to recover your bitcoins.
Someone posted another thread a few weeks ago asking if there were any other online wallet providers which offered a similar type of service and so far, Blockchain.info seems to be unique in this regard. It also explains why they are so popular:
https://bitcointalksearch.org/topic/is-blockchaininfo-the-only-online-wallet-where-you-have-control-over-your-keys-1076837There should also be a distinction between what is meant by the term "online wallet" vs. "offline wallet". On one end of the extreme, you could choose to store your coins on a shady exchange's online wallet which could shut down and disappear with your coins at any moment. Then there are the more "legitimate" online wallet services like Circle and Xapo which claim to insure your deposits in case something goes wrong. However, insurance is a legal matter and not a technical one. Insured Bitcoin wallet services are still a very new thing and it's not known whether or not they will actually pay out in the event of a massive loss. Personally, I wouldn't put much trust into these services.
Then you have Blockchain.info and multi-sig online wallets where you actually have some actual control over your coins on a technical level since you own the private key to your addresses. This is safer than the above and are probably suitable for storing medium amounts of BTC.
Moving onto offline wallets, there are lightweight wallets like Electrum and MultiBit which communicate with and depend on external servers to send and receive transactions and the default Bitcoin-Qt wallet which requires you to download the entire blockchain and be a full node. Finally, on the other extreme end of the spectrum, there are offline paper wallets. Compared with all the other options, paper wallets that have been generated in a secure manner using an offline computer are supposed to be the safest storage method and are recommended for storing larger amounts.