The best replacement for user stories is use cases.

The best alternatives to user stories are rated as following:
★★★★★  use cases
★★  tasks
★★  features
★★  functional requirements
★  scenarios
★  requirements
★  use case
  personas
  epics
  job stories
  business requirements

Your alternative for user stories is not listed? Suggest!
Speaking about user stories, one probably shouldn't forget about agile.

Other search terms related to user stories and their ratings are:
★★★★★ agile
★★★★ scrum
★★★★ acceptance criteria
You have in mind a companion for user stories and it is not yet here? Propose!



Google Trends

Scale is based on the average worldwide traffic of user stories in all years. Click on graph for more.
user stories
use cases
tasks