"Proof of Stake" Altcoins
Here are some examples of altcoins that use the Proof of Stake verification method:

Peercoin
Peercoin was the first digital currency to use the Proof of Stake method to generate coins. The mining process used to confirm Bitcoin transactions has been criticized because the specialized hardware that is now needed uses a lot of energy. It is also very expensive, which has lead to centralization. Peercoin aimed to fix this by introducing an alternative method of generating coins alongside mining. Known as "Minting", this method allows new blocks to be generated by randomly selected users who have coins in their wallet. A user's eligibility for a block reward is determined by "coin age", which is the number of coins in their wallet multiplied by the number of days they have been there. The user must have coins in the wallet for at least 30 days to be eligible for a block reward.

Vericoin
Vericoin is a Proof-of-Stake coin that introduced a few innovative and notable features. One, called VeriSMS, lets users send Vericoins to each other through text messages. Most significantly, it has a feature called "VeriBit" which allows it to be spent wherever Bitcoin is accepted. Of course, it would be easier to just spend Bitcoin, but a coin that generates interest through staking, can be counted in whole coins rather than having to be divided down to four or more decimal places, and is easily spendable could be very appealing to some.

Reddcoin
Dubbed "The Social Currency" by its developers, Reddcoin was created to be a "tipping" currency similar to Dogecoin, and integration with social networks was a big focus early on. Reddcoin also improves on the Proof of Stake system introduced by Peercoin, by introducing a revised method called Proof of Stake Velocity. A common criticism of the Proof of Stake method is that it encourages people to hoard their coins rather than spend them. Reddcoin's PoSV aimed to fix this by factoring activity between addresses into the minting process- and giving a higher reward right after they are moved. This also helps to secure the network, since a wallet has to be online to spend or move coins.