Latest Team Foundation Server and Visual Studio Team Services can generate a build status badge.
BTW, I don't find how can I link it to a public build details of latest build (or even other past builds).
For example, my build status badge is this one:
For example, I've another old project hosted on GitHub and it has CI with Travis, and I both have a build badge and a build summary: https://travis-ci.org/mfidemraizer/joopl/builds
I'm just asking for getting this summary on Visual Studio Team Services.
There is no way to do this, the Build Badge only provide an image to show the build status for now.
There are several feature requests submitted on VSTS User Voice, you can vote them: