There is a lot of free video editing software available for you. The best free video editing software for desktops, for me, would be Blender. Though, it is not primarily for video editing. But it has basic functionality with regard to video editing. It is primarily a total solution for any 3D and 2D animation needs. You can create animation with this tool. It has been described as comparable to Maya in making 3D animation. It is open-source so it is free-to-use, it is a complete program, and you have all you need to finish your project. For your video editing needs, you will be able to do basic tasks like cuts and splicing but it also allows complex tasks such as having live preview, speed control, transitions, keyframes, filters, and manipulation of layers. It also allows you to basic audio editing functions. That is a whole lot of functionality for a free-to0use program. Let us not forget that this program is continually progressing because of the updates being submitted by programmers and fellow editors that contribute to making the experience of using it better.
For mobile devices, I am torn between choosing Adobe Rush, Quik, or Promo.com. Each of these mobile applications is great for accomplishing your video editing needs while on the move. It provides very basic video editing functions but I think they are more than enough for the free-to-use tag.