Shortest Unique Substrings

Python 3, 124 123 111 96 bytes

f=lambda s,n=1:[x for x in[s[i:i+n]for i in range(len(s)+1)]if s.find(x)==s.rfind(x)]or f(s,n+1)

Looks for strings such that the first occurrence from the left is the same as the first occurrence from the right. The +1 in the range is to accommodate for the empty string case.

Now if only Python had a .count() which counted overlapping matches, then this would have been a fair bit shorter.

Mathematica, 95 94 79 bytes

Cases[Tally@StringCases[#,___,Overlaps->All],{s_,1}:>s]~MinimalBy~StringLength&

StringCases gets me all possible substrings, the Tally and Cases filter out those that appear more than once and MinimalBy finds those that are shortest.

GolfScript, 44 bytes

:S;-1:x{;S,x):x-),{S>x<}%:^1/{^\/,2=},.!}do`

Takes input as a string on stdin and outputs in a double-array syntax: e.g. [["b"] ["c"]]. Online demo

Dissection

:S;          # Store input in S and pop it
-1:x         # Store -1 in x
{            # do-while loop
  ;          #   Pop x the first time and [] every subsequent time
  S,x):x-),  #   Increment x and build an array [0 1 ... len(S)-x]
  {S>x<}%    #   Map that array to [substr(S,0,x) substr(S,1,x) ...]
  :^         #   Store in ^ (to avoid the token coalescing with the next char)
  1/         #   Split by length 1 to iterate over 1-elt arrays rather than strings
  {^\/,2=},  #   Filter to arrays which occur exactly once as a subarray of ^
  .!         #   Duplicate and test emptiness
}do          # end do-while loop: loop if the filtered array is empty
`            # Stringify for output

This is arranged such that no special case is required for the empty string (which I've included as a test case in the online demo linked above).