Slot # corresponding to subset of a list in select

Select[#, #[[1]] == 1 &] & /@ all
Select[#[[1]] == 1 &] /@ all (* thanks: Mr.Wizard *)
Pick[all, #[[1]] == 1 & /@ # & /@ all]
Pick[all, all[[All, All, 1]], 1]
Map[If[#[[1]] == 1, #, ## &[]] &, all, {2}]

all give

{{{1, 2}, {1, 3}}, {{1, 1}, {1, 5}}}

DeleteCases[all, {Except[1], _}, 2]

{{{1, 2}, {1, 3}}, {{1, 1}, {1, 5}}}

Or, more generally:

 DeleteCases[{all, all}, {Except[1], _?AtomQ}, -1] // MatrixForm

At least for the example given:

Cases[{1, _}] /@ all

{{{1, 2}, {1, 3}}, {{1, 1}, {1, 5}}}