Handheld operating systems are specifically designed to run on portable devices such as smartphones, tablets, and PDAs (Personal Digital Assistants). These operating systems are optimized for small screens, touch input, and limited hardware resources.

The correct answer is option 'C' - A PDA (Personal Digital Assistant) uses a handheld operating system.

Now let's break down each option and explain why the answer is C:

a) A supercomputer:
Supercomputers are high-performance machines used for complex scientific simulations and calculations. They typically run on specialized operating systems designed for parallel processing and massive data handling. Supercomputers do not use handheld operating systems as they are not portable devices.

b) A personal computer:
Personal computers, also known as desktop or laptop computers, run on operating systems such as Windows, macOS, or Linux. These operating systems are not specifically designed for handheld devices, although some versions may have touch-enabled features. Personal computers do not use handheld operating systems.

c) A PDA:
A PDA, or Personal Digital Assistant, is a handheld device that provides personal information management, communication, and basic computing capabilities. PDAs typically run on handheld operating systems such as Palm OS, Windows CE, or Pocket PC. These operating systems are optimized for small screens, touch input, and limited hardware resources.

d) A mainframe:
Mainframes are large, powerful computers used for processing massive amounts of data and running critical business applications. They typically run on specialized operating systems designed for high reliability, security, and scalability. Mainframes do not use handheld operating systems as they are not portable devices.

e) None of the above:
The correct answer is not "None of the above" because a PDA is a handheld device that uses a handheld operating system.

In conclusion, out of the given options, a PDA is the only device that uses a handheld operating system.