Banda de Moebius

a[s_] := {Cos[s], Sin[s], 0}; v = {0, 0, 1};
X[t_, 
          s_] := (2  t Cos[s/2] + 1 - Cos[s/2]) a[
              s] + (2  t - 1) Sin[s/2] v;
ParametricPlot3D[Evaluate[X[t, s]], {t, 0.3,  .7}, {s, 0, 2  Pi}, 
      PlotPoints -> 30]

Generalización de bandas de Moebius

Y[k_][t_, 
          s_] := (2  t Cos[k s/2] + 1 - Cos[k s/2]) a[
              s] + (2  t - 1) Sin[k s/2] v;

k = 3; 
    ParametricPlot3D[Evaluate[Y[k][t, s]], {t, 0.3,  .7}, {s, 0, 2  Pi}, 
      PlotPoints -> 50]

Una superficie parametrizada que no es superficie

alpha[t_] := {0, t^3 - 4  t, t^2 - 4};
Y[t_, s_] := alpha[t] + s {1, 0, 0}; 
ParametricPlot3D[Y[t, s], {t, -3, 3}, {s, -3, 3}, 
      ViewPoint -> {3, 2,  .5}, Compiled -> False, 
      PlotPoints -> {80, 15}]

Helicoide

beta[t_] := {Cos[t], Sin[t], t};
Z[t_, s_] := 
        beta[t] + s {Cos[t], Sin[t], 0};
t0 = 6; s0 = 10;
ParametricPlot3D[Z[t, s], {t, -t0, t0}, {s, -s0, s0}, 
      Compiled -> False, 
PlotPoints -> {60, 35}, 
      ViewPoint -> {1, 2,  .6}]

Catenoide

ParametricPlot3D[{ Cos[s] Cosh[t],   Sin[s] Cosh[t], 
        t}, {t, -2, 2}, {s, 0, 2  Pi}, ViewPoint -> {1, 2,  .6}, 
      PlotPoints -> 50]

ParametricPlot3D[
      Table[{c Cos[s] Cosh[t/c],  c Sin[s] Cosh[t/c], t}, {c,  .2, 
            1,  .2}] // Evaluate, {t, -2, 2}, {s, 0, 2  Pi}, 
      ViewPoint -> {1, 2, 0.4}, PlotPoints -> 20]