Another useful thing to do when testing such things is to determine whether packed arrays are unpacking. For all of your cases there is a lot of unpacking going on (I've only shown the first of such messages...)
In[1]:= On["Packing"]
In[2]:= test = RandomInteger[{-25, 25}, {10^6, 2}];
In[3]:= (res1 = Cases[test, {_, _?Positive}]); // AbsoluteTiming
During evaluation of In[3]:= Developer`FromPackedArray::punpackl1: Unpacking array with dimensions {1000000,2} to level 1. >>
Out[3]= {1.060814, Null}
In[4]:= (res2 = Select[test, #[[2]] > 0 &]); // AbsoluteTiming
During evaluation of In[4]:= Developer`FromPackedArray::punpackl1: Unpacking array with dimensions {1000000,2} to level 1. >>
Out[4]= {1.856424, Null}
In[5]:= (res3 =
test[[Flatten@
Position[test[[All, 2]], _?Positive]]]); // AbsoluteTiming
During evaluation of In[5]:= Developer`FromPackedArray::unpack: Unpacking array in call to Position. >>
Out[5]= {4.742461, Null}
Now Pick can be written so that it doesn't unpack...
In[6]:= (res4 = Pick[test, Sign[test[[All, 2]]], 1]); // AbsoluteTiming
Out[6]= {0.046801, Null}
In[8]:= Off["Packing"]
In[9]:= res1 == res2 == res3 == res4
Out[9]= True
