Implement a FuzzyFinder

CJam, 18 15 bytes

{1$#)}q~2$,@$p;

Try it online in the CJam interpreter.

I/O

Input:

"mig" ["imig" "mig" "migd" "do" "Mig"]

Output:

["mig" "migd" "imig"]

How it works

      q~        e# Read and evaluate the input from STDIN.
                e# Pushes a needle and an array of haystacks.
{    }          e# Define a code block:
 1$             e#   Copy the needle.
   #            e#   Compute the index of the needle in the haystack.
    )           e#   Add 1 to the index.
        2$      e# Copy the block.
          ,     e# Filter: Keep only haystacks for which the code block
                e#         pushed a non-zero value.
           @    e# Rotate the block on top of the stack.
            $   e# Sort: Arrange the haystacks according to the values
                e#       pushed by the code block.
             p  e# Print the filtered and sorted haystacks.
              ; e# Discard the needle.

Pyth, 9 bytes

oxNzf}zTQ

Try it online: Demonstration

Explanation:

            implicit: z = input string, Q = input list
    f   Q   filter Q for elements T, which satisfy:
     }zT      z is substring of T
o           order the remaining strings N by:
 xNz          the index of z in N

Python 2, 65

def f(s,l):g=lambda x:x.find(s)+1;print sorted(filter(g,l),key=g)

The expression x.find(s) returns the position of the first occurrence of s in x, giving -1 for no match. We add 1 to the result to that no-match corresponds to 0, letting us filter them out. We then sort by the match position, which is unaffected by shifting by 1.