I worked with "hollywood graded vfx houses" or what are they called these days, and they all have different workflows depending on what they are going for.
Software: Almost all of them had After effects/Nuke/Flame or even Smoke (discontinued these days) for compositing/motion graphics. Also all of them had custom made software for their pipelines. Note: While after effects is not ideal for a linear workflow, it's been used a lot for motion graphics AND for compositing in a professional workflow. But so is Nuke, Flame and Fusion.
File formats: Sometimes they used TIFF, other times they used EXR. It was, again, really dependable on what they needed it for. If they're doing a Marvel or other big budget movie, they keep the highest possible quality along the whole process. They then most likely work with what's called a scene-referred workflow, using Open EXR. For some smaller productions, with a smaller team they sometimes use log DPX. A small note: If you need speed in After effects, apparantly the fastest format is DPX. But Octane doesn't support it (yet).
That being said, for (big or small) commercials for example, I worked on projects that had to be done in a short amount of time, in After effects, using TIFF or even Png. While this is not a common file format for high-end vfx pipeline, it's not 'banished' or impossible to work with at all .
Either way, if one individual is working for the new Deadpool movie, they most likely end up being making massive amount of rock textures for months, or modelling part of a suit for Wolverine (cause they have a giant team and a person for every small job) so they don't have to worry too much about the pipeline or end file format for that matter anyway.
BUT if you're doing work in your own company, it's all about finding your own best worflow and making it look great. I can guarantee you that 95% of the work that freelancers or small teams do for corporate's that are not vfx /production houses, the client only cares about how good it looks. I never ever had any corporate CEO say: "I really like your animation, but you used PNG in your workflow. I can't stand PNG! Please do it again, but this time use EXR". Only if you're involved with other production houses, or have to work together with other pipelines, then I would say, Use a linear workflow with OpenEXR as a start, or speak with them and together set-up a pipeline that works best for the both of you. If they like to work in PNG, then so be it. I can guarantee you will not explode, nor will you ever break into pieces and be transported to another multiverse cause you used the 'wrong format'.
Cheerio!