Which software displays advertisements based on user spying?

Adware is a type of software specifically designed to display advertisements on the user's device. It often monitors user behavior and preferences, allowing it to target ads more effectively based on the data it collects. This collection of data is done typically in a non-intrusive way, but it can infringe on user privacy, as the software might track which websites a user visits, what products they look at, and how they interact with different digital content.

In contrast, worms are self-replicating malware that spread through networks without the need for user intervention, while Trojans disguise themselves as legitimate software to deceive users into executing them, often leading to malicious actions. Grayware is a term used to describe unwanted software that might not be classified as malware but still can exhibit annoying or undesirable behaviors, such as tracking user activity without clear consent.

Adware stands apart in its primary function, focusing specifically on serving ads based on gathered user data, which is why it is the correct choice in this context.