Code Golf: Your own pet ASCII snake

05AB1E, 15 14 bytes

30DF2Ý<+ΩD0sú,

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Uses 0.

Explanation

30DF2Ý<+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] ...
      <         # 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

Random Brainfuck, 123 122 121 bytes

+[--[<]>>+<-]>+[->+>+>+<<<]++++++++++>>++<[>>[->+<<.>]>[-<+>]>?>+++<[>->+<[>]>[<+>-]<<[<]>-]>-->,<[-<<<+>>>]<<<<+.-<<.>-]

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Random Brainfuck is an extension of brainfuck, with the helpful addition of the ? command, which sets the current cell to a random byte. This prints a snake made of !s, which looks more like footsteps than a snake funnily enough.

How It Works:

+[--[<]>>+<-]>+ Create the value 30
[->+>+>+<<<]    Copy it three times
++++++++++      Create a newline cell
>>++<            Adds 2 to the second copy to make it a space and move to the counter
[ While counter
  >>[->+<<.>]>[-<+>] Print out the padding cell number of spaces
  ?>+++<[>->+<[>]>[<+>-]<<[<]>-] Get 3-(random byte%3)
  >-->,<[-<<<+>>>]   Add (result-2) to the padding cell
  <<<<+.-<           Print an exclamation mark
  <<.                Print a newline
  >-                 Decrement counter
] end loop

Another solution that sticks to the letter of the question, rather than the spirit.

87 bytes

+[--[<]>>+<-]>+[->+>+>+<<<]++++++++++>++>[>[->+<<<.>>]>[-<+>]?[,<+>]?[,<->]<<<+.-<.>>-]

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This one is heavily biased towards leaving the padding alone, but increasing or decreasing the padding are both equally possible. Each one has a slightly less than 1 in 256 chance to happen.

C (gcc), 61 58 56 bytes

Answer edited to reflect rules changes...

i;f(s){for(s=i=31;--i;printf("%*d\n",s+=1-rand()%3,8));}

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