Ruby, 63

p=->(w,l){l.find_all{|x|x[w]}.sort{|a,b|a.index(w)-b.index(w)}}

Run

irb(main):022:0> p["mig", ["imig", "mig", "migd", "do", "Mig"]]
=> ["migd", "mig", "imig"]

Notes

1. First find all matching words with find_all
2. Sort by index position of word to be searched.

Edit (by daneiro)

Ruby, 49

p=->w,l{l.select{|x|x[w]}.sort_by{|e|e.index(w)}}

Haskell, 129 116

116 (Thanks to Franky):

import Data.List
h s=map snd.sort.map(\x->((head[c|c<-[0..length x],isPrefixOf s(drop c x)]),x)).filter(isInfixOf s)

129:

import Data.List
f s n=map snd(sort(map(\x->((head [c|c<-[0..length x],isPrefixOf s(drop c x)]),x))(filter(\x->isInfixOf s x)n)))

Well, it's pretty long, maybe I'll find how to short it up a bit...

ORACLE, 60

Does this count?

select * from t where a like '%mig%' order by instr(a,'mig')

GolfScript, 13 bytes

~{?)}+\1$,\$`

This is one of those rare occasions where GolfScript can beat CJam, by using block concatenation and taking a few liberties with the input which can be formatted however you wish.

Try it online in Web GolfScript.

I/O

Input

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

Output

["migd" "mig" "imig"]

How it works

~             # Evaluate the input from STDIN.
              # Pushes an array of haystacks and a needle in a block.
 {?)}         # Push a code block that computes an index and increments it.
     +        # Concatenate that block with the needle block.
      \1$     # Swap the block with the arrays of haystacks and copy the block.
         ,    # Filter: Keep only haystacks for which the code block
              #         pushed a non-zero value.
          \   # Swap the array of haystacks with the code block.
           $  # Sort: Arrange the haystacks according to the values
              #       pushed by the code block.
            ` # Inspect: Format the array for pretty printing.

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.

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.

Python 2, 69 68 Bytes

I just created a function which takes s as the string to match in a list of strings n

Edit 1: Thanks to Jakube for golfing off a byte.

f=lambda s,n:sorted([x for x in n if s in x],key=lambda x:x.find(s))

Check it out here.

[Hold] Pyth, 24 Bytes

JwKcwdVlK=G.)KI}JGaYG))Y

Try is here

I'm pretty new at Code Golfing/Pyth so I'm not sure it's optimal, but I'm working on it!

Update: I don't think I'm actually sorting correctly, and I can't seem to get it to work. I know that o is order-by, and I need to sort by position of S so I am using .:GlJ in order to find all the substrings of the length of S for the current element G and then x in order to find the index of the first occurrence of S, but I can't seem to set lambda correctly.

JavaScript ES6, 68 bytes

(s,l,f=j=>j.indexOf(s))=>l.filter(w=>~f(w)).sort((a,b)=>f(a)>f(b))

This is an anonymous function that takes parameters s (file path string) and l (array of strings). The Stack Snippet below contains ungolfed code converted to ES5 so more people can test it easily. (If you have Firefox, you can use edc65's prettier test suite found in his answer.)

f=function(s,l){
  g=function(j){
    return j.search(s)
  }
  
  return l.filter(function(w){
    return ~g(w)
  }).sort(function(a,b){
    return g(a)>g(b)
  })
}

id=document.getElementById;run=function(){document.getElementById('output').innerHTML=f(document.getElementById('s').value,document.getElementById('l').value.split(', ')).join(', ')};document.getElementById('run').onclick=run;run()

<label>File path: <input type="text" id="s" value="mig" /></label><br />
<label>Files: <input type="text" id="l" value="imig, mig, migd, do, Mig" /></label><br />
<button id="run">Run</button><br />
Output: <output id="output"></output>

JavaScript (ES6), 68

That's almost the same of @NBM answer (even if it's not copied), so I don't expect upvotes. Enjoy the snippet anyway

A function with a string and a string array arguments, return a string array. Filter then sort.

Test runnign the snippet below (being EcmaScript 6, Firefox only)

f=(s,l,i=t=>t.indexOf(s))=>l.filter(t=>~i(t)).sort((t,u)=>i(t)-i(u))

$(function(){
  $("#S,#L").on("keyup", 
   function() { 
     $('#O').val(f(S.value,L.value.split('\n')).join('\n'))
   } );
  $("#S").trigger('keyup');
})

#S,#L,#O { width: 400px }
#L,#O { height: 100px }

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
Input String<br><input id=S value='mig'><br>
Input List<br><textarea id=L>
imig
mig
migd
do
Mig
</textarea><br>
Output List<br><textarea id=O readonly></textarea>