Several parallel operations become unreadable in status bar
Pan News Reader has a button with the Number of tasks as a label and is followed by a line describing the currently processed task.
If you click on the Tasks button you get a window where you can manage the task (cancel, reorder, pause).
Evolution tasks are no longer filling my Evolution status bar most of the time as now the bandwidth and processing are way faster, though it has been years since I postponed requesting this feature. I find it way easier than a status bar where only "cancel" buttons are left visible because there are so many labels that do not fit anymore (and also they constantly move on the status bar which is distracting and can be uneasy to click on while they move because other tasks have completed or other have been added to the queue).
As a first step, does such a design make sense for Evolution or Gnome HIG?
The only downside of the Pan design is that with Gnome Shell opening windows in the background when you click the Task button you then add to switch windows to view the Tasks window. So there might be way to have Gnome Shell open subwindows up the stack without making them modal (or make them modal, but if one want to spend his time tweaking the task list it might be a problem to not be able to read emails at the same time).