Edit: Wrong, wrong, see my edit below. This describes a flying, spinning, vibrating dumbbell-shaped object. (Removed reference to oscillator so not to confuse others.)

It has 6 degrees of freedom:

1. Translation, x direction

2. Translation, y direction

3. Translation, z direction

4. Rotational, about x axis

5. Rotational, about y axis

6. Vibrational, z axis

We've oriented it along the Z axis so that we can discount rotational degree about same axis, and so that we only have to consider the one vibrational degree.

Edit: I've confused this with another. I came back to correct it and saw someone else had done so for me.

Again, to really simplify it, consider the oscillator's "energy" in one dimension goes like: E = 1/2mv^2 + 1/2kx, where x is displacement. Two degrees here, and three dimensions, for six degrees. See other answers for a more complete explanation.