Flatten[list, {{s11, s12, ...}, {s21, s22, ...}, ...}]

In[63]:=  Flatten[{{1,2,3},{4,5},{6,7},{8,9,10}},{{2},{1}}]
Out[63]= {{1,4,6,8},{2,5,7,9},{3,10}}

In[65]:= Flatten[{{1,4,6,8},{2,5,7,9},{3,10}},{{2},{1}}]
Out[65]= {{1,2,3},{4,5,10},{6,7},{8,9}}

In[67]:= Flatten/@Flatten[{{1,4,6,8},{2,5,7,9},{3,{},{},10}},{{2},{1}}]
Out[67]= {{1,2,3},{4,5},{6,7},{8,9,10}}

In[68]:= Flatten[{{{1,2,3},{4,5}},{{6,7},{8,9,10}}},{{2},{1},{3}}]
Out[68]= {{{1,2,3},{6,7}},{{4,5},{8,9,10}}}

In[69]:=  Flatten[{{{1, 2, 3}, {4, 5}}, {{6, 7}, {8, 9,  10}}}, {{3}, {1}, {2}}]
Out[69]= {{{1, 4}, {6, 8}}, {{2, 5}, {7, 9}}, {{3}, {10}}}

In[72]:=  step1 = Flatten[{{{1,2,3},{4,5}},{{6,7},{8,9,10}}},{{1},{3},{2}}]
Out[72]= {{{1,4},{2,5},{3}},{{6,8},{7,9},{10}}}

In[73]:= step2 =  Flatten[step1,{{2},{1},{3}}]
Out[73]= {{{1,4},{6,8}},{{2,5},{7,9}},{{3},{10}}}

Flatten[{{{111, 112, 113}, {121, 122}}, {{211, 212}, {221, 222, 223}}}, {{3}, {1}, {2}}]
(*
==> {{{111, 121}, {211, 221}}, {{112, 122}, {212, 222}}, {{113}, {223}}}
*)

In[74]:=  Flatten[{{{1,2,3},{4,5}},{{6,7},{8,9,10}}},{{2},{1,3}}]
Out[74]= {{1,2,3,6,7},{4,5,8,9,10}}

$m = Array[Subscript[m, Row[{##}]]&, {4, 3, 2}];
$m // MatrixForm

$m === Table[$m[[i, j, k]], {i, 1, 4}, {j, 1, 3}, {k, 1, 2}]
(* True *)

$r = Table[$m[[i, j, k]], {j, 1, 3}, {i, 1, 4}, {k, 1, 2}];
$r // MatrixForm

$r === Flatten[$m, {{2}, {1}, {3}}]
(* True *)

$m === Flatten[$m, {{1}, {2}, {3}}]
(* True *)

Flatten[$m, {{3}, {2}, {1}}] // MatrixForm

Flatten[$m, {{3}}] === Flatten[$m, {{3}, {1}, {2}}]
(* True *)

Flatten[$m, {3}] === Flatten[$m, {{3}}]
(* True *)

$m === Flatten[$m, {}] === Flatten[$m, {1}] === Flatten[$m, {{1}, {2}, {3}}]
(* True *)

Flatten[$m, {{1}, {2}, {3}}] // MatrixForm

Flatten[$m, {{1, 2}, {3}}] // MatrixForm

Flatten[$m, {{1}, {2, 3}}] // MatrixForm

Flatten[$m, {{1, 2, 3}}]

Flatten[$m, {{1, 2, 3}}] === Flatten[$m, {1, 2, 3}] === Flatten[$m]
(* True *)

$n = Array[n[##]&, {2, 2, 2, 2, 2}];

Flatten[$n, {{1}, {2}, {3}, {4}, {5}}] === Flatten[$n, 0]
(* True *)

Flatten[$n, {{1, 2}, {3}, {4}, {5}}] === Flatten[$n, 1]
(* True *)

Flatten[$n, {{1, 2, 3}, {4}, {5}}] === Flatten[$n, 2]
(* True *)

Flatten[$n, {{1, 2, 3, 4}, {5}}] === Flatten[$n, 3]
(* True *)

$t = Array[# Range[#]&, {5}];
$t // TableForm
(*
1
2   4
3   6   9
4   8   12  16
5   10  15  20  25
*)

Flatten[$t, {{2}, {1}}] // TableForm
(*
1   2   3   4   5
4   6   8   10
9   12  15
16  20
25
*)

Flatten[$t, {{2, 1}}]
(* {1,2,3,4,5,4,6,8,10,9,12,15,16,20,25} *)

lists = {
   {{{1, 0}, {1, 1}}, {{2, 0}, {2, 4}}, {{3, 0}}},
   {{{1, 2}, {1, 3}}, {{2, Sqrt[2]}}, {{3, 4}}}
   (*, more lists... *)
   };

list2 = {
  {{1, 0}, {1, 1}, {2, 0}, {2, 4}, {3, 0}},
  {{1, 2}, {1, 3}, {2, Sqrt[2]}, {3, 4}}
  (*, more lists... *)
  }

list2 = Flatten[lists, {{1}, {2, 3}}];

TreeForm@{{1,2},{3,4}}

{<|"index" -> {1, 1}, "value" -> 1|>, <|"index" -> {1, 2}, "value" -> 2|>, <|"index" -> {2, 1}, "value" -> 3|>, <|"index" -> {2, 2}, "value" -> 4|>}

Flatten[mat,{{2},{1}}]

Flatten[mat,{{1,2}}]

{{{{1,2},{5,6}},{{3,4},{7,8}}},{{{9,10},{13,14}},{{11,12},{15,16}}}}

Flatten[mat,{{1,3},{2,4}}]

img=Image@RandomReal[1,{10,10}]

ConstantArray[ImageData@img,{10,10}]

Image@Flatten[ConstantArray[ImageData@img,{10,10}],{{3,1},{4,2}}]