  We prove that knot Floer homology and Khovanov homology can detect
non-fibered knots, including the knot $5_2$, and that HOMFLY homology detects
infinitely many such knots. This is the first time that a Floer theory or any
of the Khovanov-Rozansky link homology theories has been shown to detect
non-fibered knots, or infinitely many knots. These results rely on our main
theorem, which gives a complete classification of genus-1 knots in $S^3$ whose
knot Floer homology in the top Alexander grading is 2-dimensional.
