I am getting a 404 on http://localhost/wp-json/wp/v2/posts route. I changed permalink type to plain. Still got the same issue.
If its useful, I am ubuntu 16.04 desktop recent upgrade from 15.10, suffered issues with php5.6 to 7.0 upgrade. Am now on php 7.0
Edit1: Tried every other permalink setting combination, no results!
http://localhost/?rest_route=/wp/v2/posts
should works.
Refer https://developer.wordpress.org/rest-api/using-the-rest-api/discovery/ for details.