Software as a service (SaaS) is a method of providing software as a service through the Internet. Instead of installing and maintaining software, you just use the Internet to access it, eliminating the need for complicated software and device maintenance.
Ex: Amazon EC2 fall under SaaS
Platform as a service (PaaS) is a cloud computing concept in which consumers get hardware and software tools from a third-party supplier through the internet. These tools are typically required for application development. The hardware and software are hosted on the PaaS provider's own infrastructure.
Examples: AWS Elastic Beanstalk, Windows Azure.
Infrastructure as a service (IaaS) is a standardized, highly automated product in which a service provider's computer resources, supported by storage and networking capabilities, are made available on demand to consumers. In near real time, resources are scalable and elastic, and usage is metered.
Examples: Google Apps, Microsoft Office 365.
In terms of popularity, all of these services are quite popular. It's only a matter of selecting one best suits your needs. For example, if you want to set up a Hadoop cluster on which to execute MapReduce processes, EC2, which is IaaS, is a wonderful choice. On the other hand, if you have an application developed in a certain language that you wish to deploy in the cloud, you would use a PaaS platform like Heroku.