Waifu2x is a utility that has utilized machine learning algorithms to analyze anime style art in order to learn how to "essentially" re-draw image at a higher resolution. It's not just a simple image sharpening filter.

I'm fairly sure waifu2x is the only publicly available utility that uses such a method to process images. So for most anime style art, you won't find anything better.

However, art styles vary and some images just don't turn out well with waifu2x. Especially images with grainy textures or a dense field of stars. In such cases, you may find some combination of standard image filters that works better.

But when fine details such as lacework are downscaled so far that the details are no longer clear enough to recognize, nothing can fix that (except an artist)...