I'm also looking for some free software (on Windows, Linux or Android) or Photoshop plugin but the only ones I found that are free are Matlab programs. Those cannot run alone unless you converted the Matlab files to executable files.
We have developed a Matlab program with graphical user interface that implements several image super-resolution methods developed in this project.
This application is availabe for the use of researchers and companies.

Super-Resolution is a Matlab program with graphical user interface that implements several image registration and reconstruction algorithms for super-resolution imaging.
This program is distributed under the General Public Licence GPL, which is included in the GPL file with the code.
Most remaining softwares I found are for videos since each frame only differs a tiny amount and the software can combine multiple consecutive images into a sharper video or snapshot of the video.
This is a Matlab-based software package for resolution enhancement from video, developed at the Multi-Dimensional Signal Processing (MDSP) research lab at the University of California at Santa Cruz, led by Peyman Milanfar.
The main objective of this software tool is the implementation of several superresolution techniques. In particular, the techniques described in 1, [2], [3], and several references therein are included. The techniques implemented cover robust methods, dynamic color superresolution methods, and simultaneous demosaicing and resolution enhancement.
Some specific features of the software package are:
- As part of this software package, motion estimation is done automatically by the program, or independently estimated motion vectors may be provided by the user.
- The user is able to specify the region of interest to be processed.
- A basic tracking algorithm is incorporated in the program so that if only a certain part of the input images are important for the user (a car moving in a crowded street), this region can be tracked and another data sequence containing only that particular object is produced.
- The parameters of the imaging system (such as the point-spread function) may be specified by the user.
- The input image files may be given as .mat (Matlab data file) or .avi format.
- The output generated by the program can be .mat (Matlab data file) or .avi format.
- Producing color or grayscale output images are optional, given color input frames.
- For purposes of experimentation, the software package is capable of producing simulated video data for different imaging scenarios from a single high resolution input image, with user-controlled parameters.

This trial-ware uses the same technique to upscale video resolution. You may extract the frame later, after up-scaling easily.
Video upsizing plug-in for Adobe After Effects 6.5+ and Adobe Premiere Pro 2.0. Uses information from neighbor frames to upsize current frame. It provides more details than any known upsize method. Now you can convert SD video to HD!
The plugin contains two modes of super-resolution and a Lanczos resize method which can serve as a fast high-quality replacement for bicubic resize. All modes give much sharper and detailed video than built-in scale transform.
The upsize method used in this plugin works only with progressive video (not interlaced). The plugin can also be used to downsize video with good quality.
You may use this free trial as long as you want. Unregistered version adds watermark over the video. Register to turn off the watermark.
Another similar software is vReveal but it seems the main page has been down


Video upsizing plug-in for Adobe After Effects and Premiere Pro (CS5, CS6, CC 2014 - 2015). Uses information from neighbor frames to upsize current frame. It provides more details than any known upsize method. Now you can convert SD video to HD! The plugin contains two modes of super-resolution and a Lanczos resize method which can serve as a fast high-quality replacement for bicubic resize. All modes give much sharper and detailed video than built-in scale transform. The upsize method used in this plugin works only with progressive video (not interlaced). The plugin can also be used to downsize video with good quality.
You may use this free trial as long as you want. Unregistered version adds watermark over the video. Register to turn off the watermark.
SuperResolution plugin is a free trial software application from the Video Tools subcategory, part of the Audio & Multimedia category. The app is currently available in English and it was last updated on 2016-08-11. The program can be installed on Windows 7, Windows8, WinVista, WinVista x64, WinXP, Win10 x64, Win8 x64, Windows 8, Windows 10.
A similar question on photography.SE: Is there a photoshop plugin or any other piece of software that can do super resolution?