3 added 26 characters in body
source | link

How about

r[x_] := 1/((x - 2) Sqrt[x - 1])

Select[x /. Solve[Denominator[r[x]] == 0, x],
       SeriesCoefficient[r[x], {x, #, -1}] != 0 &]
  (* {2} *)

Or a more direct solution (with thanks to xzczd):

Select[x /. Solve[Denominator[r[x]] == 0, x], 
 Head@Residue[r[x]      FreeQ[Residue[r[x], {x, #}] =!=, Residue_Residue] &]

How about

r[x_] := 1/((x - 2) Sqrt[x - 1])

Select[x /. Solve[Denominator[r[x]] == 0, x],
       SeriesCoefficient[r[x], {x, #, -1}] != 0 &]
  (* {2} *)

Or a more direct solution:

Select[x /. Solve[Denominator[r[x]] == 0, x], Head@Residue[r[x], {x, #}] =!= Residue &]

How about

r[x_] := 1/((x - 2) Sqrt[x - 1])

Select[x /. Solve[Denominator[r[x]] == 0, x],
       SeriesCoefficient[r[x], {x, #, -1}] != 0 &]
   {2}

Or a more direct solution (with thanks to xzczd):

Select[x /. Solve[Denominator[r[x]] == 0, x], 
       FreeQ[Residue[r[x], {x, #}], _Residue] &]
2 added 134 characters in body
source | link

How about

r[x_] := 1/((x - 2) Sqrt[x - 1])

Select[x /. Solve[Denominator[r[x]] == 0, x],
       SeriesCoefficient[r[x], {x, #, -1}] != 0 &]
  (* {2} *)

Or a more direct solution:

Select[x /. Solve[Denominator[r[x]] == 0, x], Head@Residue[r[x], {x, #}] =!= Residue &]

How about

r[x_] := 1/((x - 2) Sqrt[x - 1])

Select[x /. Solve[Denominator[r[x]] == 0, x],
       SeriesCoefficient[r[x], {x, #, -1}] != 0 &]
   {2}

How about

r[x_] := 1/((x - 2) Sqrt[x - 1])

Select[x /. Solve[Denominator[r[x]] == 0, x],
       SeriesCoefficient[r[x], {x, #, -1}] != 0 &]
  (* {2} *)

Or a more direct solution:

Select[x /. Solve[Denominator[r[x]] == 0, x], Head@Residue[r[x], {x, #}] =!= Residue &]
1
source | link

How about

r[x_] := 1/((x - 2) Sqrt[x - 1])

Select[x /. Solve[Denominator[r[x]] == 0, x],
       SeriesCoefficient[r[x], {x, #, -1}] != 0 &]
   {2}
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