Code Golf: Your own pet ASCII snake
05AB1E, 15 14 bytes
30DF2Ý&lt;+ΩD0sú,

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Uses 0.
Explanation
30DF2Ý&lt;+ΩD0sú,
30D            # Push 30 to the stack (One for the amount of iterations we are going to perform and one for the initial padding)
   F           # Pop one of the 30s and perform the following that many times...
    2Ý          # Push [0,1,2] ...
      &lt;         # and create [-1,0,1] from that
       +        # Add the last padding to every entry (e.g. 30 in the beginning resulting in [29,30,31]
        Ω       # Pick one of the results at random ...
         D      # and push it to the stack twice
          0     # Push 0 (Any character will work fine here) ...
           sú   # and pad it with the randomly chosen amount of spaces in the front
             ,  # Finally print the result with a trailing newline