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Control the Location of Streamlines
Use StreamPoints to give seed points for streamlines.
In[1]:=

Click for copyable input
pts = Flatten[

   Table[{{x, y}, Hue[Random[]]}, {x, -1, 1, 0.25}, {y, -1, 1, 

     0.25}], 1];
In[2]:=

Click for copyable input
 StreamPlot[{-y^2, Sin[x]}, {x, -1, 1}, {y, -1, 1}, 

 AspectRatio -> Automatic, 

 StreamPoints -> {pts, Automatic, Scaled[2.3]}, StreamScale -> None, 

 Epilog -> {Blue, PointSize[Medium], Point[pts[[All, 1]]]}, 

 ImageSize -> Large]
Out[2]=