>  Do the speckes disappear if you specify an ior or reflection for the 
> objects?

It looks like the speckles are there even with ior or reflection or with
max_trace_level boosted.

>  Could you build a minimal scene which demonstrates the problem?

Sure, this shadow looks banded by coincident surfaces in 3.7
but not in 3.6

#declare testmat = material { texture {
      pigment {color <1,0,0> transmit 0.5}
      // finish { reflection 0.1 }
   }
   // interior{ior 1.1}
};
camera { location  <1.0, 2.0, -4.0>
   direction 1.5*z
   right     x*image_width/image_height
   look_at   <1.0, -0.2,  0.0>
   angle 30 }
light_source {<-10, 10, -5> color rgb <1, 1, 1>}
plane {y,-1 pigment{rgb <1,1,1>}}
union {
   sphere {<0.0,0,0>,0.5 material{testmat}}
   sphere {<0.1,0,0>,0.5 material{testmat}}
   sphere {<0.2,0,0>,0.5 material{testmat}}
   sphere {<0.3,0,0>,0.5 material{testmat}}
   sphere {<0.4,0,0>,0.5 material{testmat}}
   sphere {<0.5,0,0>,0.5 material{testmat}}
}

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