Virtual hosted mining
Amazon’s EC2 platform

Amazon’s Elastic Cloud Computing (EC2) platform was once a popular way of mining scrypt-based currencies, such as litecoin, feathercoin and dogecoin. Amazon comes with added costs either in the form of an advance payment for a year’s service or a ‘pay-as-you-go’ model.

However, a potentially more economical model is the ‘spot instance”, whereby users can bid for computing time at a price they are willing to pay. When Amazon has spare capacity, it will grant computing power to the highest bidder. However, if you bid too low, your mining system may never come online at all. Furthermore, you may select instances in different geographic regions that can have an effect on price.

The best type of EC2 virtual machine for mining will have virtual graphics cards. But, these are harder to access on spot instances.

At one time, it was profitable to mine litecoin this way, but not currently.
Digital ocean

This VPS service offers a very simple ‘pay-as-you-go’ service that became popular during the early days of primecoin. However, the cheapest plans offer only modest computing power. For example, the $5/month tarriff has but one core and gives single-digit kilohashes per second for scrypt currencies. It should be noted, though, that this could be cost effective for completely new currencies.
The rest

In principal, mining can be done on any type of virtual host service. However, one needs to carefully consider the computing capability on offer. While it is theoretically possible to install mining software on a web server, it would probably not be profitable.

There have been issues with hacker-controlled botnets that have been able to direct a vast amount of computing power at mining cryptocurrencies, or by developing different types of malware to infect more machines.