Valid snakes on a plane
Pyth, 22 20 bytes

ql{m+=Z*=T^.j)hCdzlz


Try it yourself or run the testsuite.

Note the ASCII values of SRL, respectively 83, 76, 82. I abuse the fact that:


  i 83 + 1 = 1
  i 76 + 1 = i
  i 82 + 1 = -i


From here I just keep a variable for the current position and current direction. For every character I multiply the current direction with the above complex number, then add it to the current position.

At the end I check if all the visited positions are unique.