Leaving the Nest

Mathematica, 56 54 52 bytes

_{-2 bytes due to Alephalpha.}

_{-2 bytes due to CatsAreFluffy.}

Cases[#,_?AtomQ,{i}]~Table~{i,Depth@#}/.{}->Nothing&

Actually deletes empty levels.

Pyth, 17

 us-GeaYsI#GQ)S#Y

The leading space is important. This filters the list on whether the values are invariant on the s function, then removes these values from the list and flatten it one level. The values are also stored in Y and when we print we remove the empty values by filtering if the sorted value of the list is truthy.

Test Suite

Alternatively, a 15 byte answer with a dubious output format:

 us-GpWJsI#GJQ)

Test Suite

Expansion:

 us-GeaYsI#GQ)S#Y     ##   Q = eval(input)
 u          Q)        ##   reduce to fixed point, starting with G = Q
        sI#G          ##   get the values that are not lists from G
                      ##   this works because s<int> = <int> but s<list> = flatter list
      aY              ##   append the list of these values to Y
     e                ##   flatten the list
   -G                 ##   remove the values in the list from G
              S#Y     ##   remove empty lists from Y

Python 2, 78 bytes

f=lambda l:l and zip(*[[x]for x in l if[]>x])+f(sum([x for x in l if[]<x],[]))