With a large share of queries coming up from mobile phones, Google is making sure that it lists websites which are mobile friendly in the search results. So it is up to the webmaster to make sure that the website is mobile friendly.
Google has now launched a new tool called Mobile friendly testing tool to help webmasters check if their websites are made mobile friendly. The tool is available here.
This report can also accessed from Google Webmasters central. You can access the mobile usability report here and also check for any errors. The updated tool provides us with room to continue to improve on its functionality, and over time.
This tool can also be accessed from mobile phones, so in case you want to check it from there, it works perfectly well.