I am learning about the wave front set of a distribution but am having difficulty understanding some details, which to me seem counter intuitive. We know the fourier transform of a smooth function decays rapidly, in fact the fourier transform of a distribution is rapidly decaying if and only if the distribution is induced by a smooth function. So we take a distribution $u$ and localize it in space by multiplying by a bump function $\phi$ with support around some $x$. We can now check the directions (or rather codirections?) where $\widehat {\phi u}$ fails to be rapidly decreasing and this gives us our wave front set (doing the same for each $x$ in the singular support of $u$.)
Here is where I get confused. It turns out that finding where $\widehat {\phi u}$ fails to be rapidly decreasing IS NOT equivalent to finding the directions along which $u$ is not smooth.
To be more specific I give an example. Consider the step function on the plane: $u=1$ on the right half and $u=0$ on the left half. The singularities occur on the vertical axis (ie the step). If we take a point on the vertical axis, the only directions in which $u$ is not singular are the vertical directions (ie tangent to the step). So I would have thought that the wave front set would consist of the vertical x axis along with all directions NOT tangent to the vertical axis. It turns out though, I have computed the Wave front set using the above definition, we get that the wave front set consists ONLY of directions normal to the step (ie horizontal directions for a step on the vertical axis). That is $\widehat {\phi u}$ is rapidly decaying for all directions except normal directions.
It seems obvious to me that the normal directions would be included and that the tangent directions are not. But my confusion is about the in between directions. If you are travelling along any direction not tangent to the step you will encounter a jump so the step function is singular for example at a 45 degree direction.
So my question is: What is happening in these in between directions that even though u is not smooth in those directions, they do not make up the wave front set. Or in another form.. what is the wave front set? because clearly 'the directions where u fails to be smooth' is not accurate as noted in the above example. I am not sure if it has something to do with the localizing of $u$ using $\phi u$.. or if perhaps my problem is confusing the notion of directions and codirections.
Out of curiosity, I computed the wave front set of $u=1$ on upper right quadrant and $u=0$ elsewhere ie this time there is a step on a corner. I found that on this corner all directions are 'bad' so they are included in the wave front set. This is what I expected, as opposed to the simple step (with no corner).