Xx[s_, t_] := s + t; Yy[s_, t_] :=  
      s - t;

Show[
Graphics[{
{RGBColor[0,  .5, 
              0], Thickness[0.005], 
            Table[Line[{Xx[s, #], Yy[s, #]} & /@ 
                  Range[-10, 10,  .01]], {s, -10, 
                10,  .5}]}, 
{Blue, Thickness[0.005], 
            Table[Line[{Xx[#, t], Yy[#, t]} & /@ 
                  Range[-10, 10,  .01]], {t, -10, 
                10,  .5}]}
}], 
PlotRange -> {{-4, 4}, {-4, 4}}, 
AspectRatio -> 
        Automatic, Frame -> True
]

Xx[s_, t_] := 2 s - 3 t ; Yy[s_, 
        t_] :=   .5 s + 2  t;

Show[
Graphics[{
{RGBColor[0,  .5, 
              0], Thickness[0.005], 
            Table[Line[{Xx[s, #], Yy[s, #]} & /@ 
                  Range[-10, 10,  .01]], {s, -10, 
                10,  .5}]}, 
{Blue, Thickness[0.005], 
            Table[Line[{Xx[#, t], Yy[#, t]} & /@ 
                  Range[-10, 10,  .01]], {t, -10, 
                10,  .5}]}
}], 
PlotRange -> {{-4, 4}, {-4, 4}}, 
AspectRatio -> 
        Automatic, Frame -> True
]

Xx[s_, t_] := (s^2) - (t^2) ; Yy[s_, t_] :=  
        2 s t ;

Show[
Graphics[{
{RGBColor[0,  .5, 
              0], Thickness[0.005], 
            Table[Line[{Xx[s, #], Yy[s, #]} & /@ 
                  Range[-10, 10,  .01]], {s, -10, 
                10,  .5}]}, 
{Blue, Thickness[0.005], 
            Table[Line[{Xx[#, t], Yy[#, t]} & /@ 
                  Range[-10, 10,  .01]], {t, -10, 
                10,  .5}]}
}], 
PlotRange -> {{-4, 4}, {-4, 4}}, 
AspectRatio -> 
        Automatic, Frame -> True
]

Xx[s_, t_] := Exp[s] Cos[t] ; Yy[s_, t_] :=  
        Exp[s] Sin[t] ;

Show[
Graphics[{
{RGBColor[0,  .5, 
              0], Thickness[0.005], 
            Table[Line[{Xx[s, #], Yy[s, #]} & /@ 
                  Range[-10, 10,  .01]], {s, -1, 
                5,  .5}]}, 
{Blue, Thickness[0.005], 
            Table[Line[{Xx[#, t], Yy[#, t]} & /@ 
                  Range[-10, 10,  .01]], {t, 0, 2 Pi, 
                (Pi/6)}]}
}], 
PlotRange -> {{-4, 4}, {-4, 4}}, 
AspectRatio -> 
        Automatic, Frame -> True
]

Xx[s_, t_] := 
        (s^3) - 3 s (t^2) ; Yy[s_, t_] :=  
        3 (s^2) t - (t^3) ;

Show[
Graphics[{
{RGBColor[0,  .5, 
              0], Thickness[0.005], 
            Table[Line[{Xx[s, #], Yy[s, #]} & /@ 
                  Range[-10, 10,  .01]], {s, -10, 
                10,  .5}]}, 
{Blue, Thickness[0.005], 
            Table[Line[{Xx[#, t], Yy[#, t]} & /@ 
                  Range[-10, 10,  .01]], {t, -10, 
                10,  .5}]}
}], 
PlotRange -> {{-4, 4}, {-4, 4}}, 
AspectRatio -> 
        Automatic, Frame -> True
]