Let $X$ be a $C^\infty$-manifold, $Y$ be its $C^\infty$-submanifold. Hörmander defines the set $I^m(X,Y)$ of conormal distributions as the set of all $u \in \mathscr D'(X)$ such that
$$
 L\_1 \ldots L\_N u \in {}^{\infty}H^{\mathrm{loc}}\_{(-m-n/4)}(X), \quad n = \dim X,
$$
for any smooth vector fields $L\_1$, $\ldots$, $L\_N$ on $X$ that are tangent to $Y$ and for any $N$, where ${}^\infty H^{\mathrm{loc}}\_{(-m-n/4)}(X)$ is the Besov space.


Is there some practical way to check whether some distribution $u \in \mathscr D'(X)$ is in fact from $I^m(X,Y)$ or I should always check the above inclusions? What are the examples of the most simple nontrivial conormal distributions for general $X$ and $Y$? If I take, e.g. $Y = f^{-1}(0)$ for some smooth submersion $f \colon X \to \mathbb R$ and take some $C^\infty$-density $\mu$ on $Y$ and then define for any $v \in C^\infty\_c(X)$ $\langle \delta\_\mu, v \rangle = \int\_Y v\mu$ is it possible to say that $\delta\_\mu \in I^m(X,Y)$ for some $m$? 



Practical way to check whether a distribution is conormal
Let us take a look at the case where $Y\equiv x\_n=0$ in $\mathbb R^n$. 


Let $a(\underbrace{\overbrace{x\_1,\dots,x\_{n-1}}^{x'},x\_n}\_x; \xi\_n)$ be a symbol in $S^m(\mathbb R^n\times \mathbb R)$ and let us consider the following "oscillatory integral":
$$
u(x)=\int\_{\mathbb R} e^{ix\_n\cdot \xi\_n} a(x,\xi\_n) d\xi\_n.
$$
This integral is not absolutely converging and we could define it as a distribution. Note that we may assume that $a$ is supported where $\vert \xi\_n\vert\ge 1$ since the contribution of a compactly supported $a$ is a smooth function. Then we have for $\phi$ in the Schwartz space, $l\in 2\mathbb N$,
formally integrating by parts,
\begin{align\*}
\langle u,\phi\rangle\_{\mathscr S',\mathscr S}&=\int\_{\mathbb R^n\times\mathbb R}
\vert \xi\_n\vert^{-l} \vert D\_{x\_n}\vert^l (e^{ix\_n\cdot \xi\_n}) a(x,\xi\_n)\phi(x) dx d\xi\_n\\
&=\sum\_{\alpha+\beta=l}c\_{\alpha\beta}\int\_{\mathbb R^n\times\mathbb R}
e^{ix\_n\cdot \xi\_n} \vert \xi\_n\vert^{-l} (D\_{x\_n}^\alpha a)(x,\xi\_n)(D\_{x\_n}^\beta\phi)(x) dx d\xi\_n,
\end{align\*}
which makes sense for $l>1+m$. We can take the last integral as the definition of the duality product.
We note also that $u$
can easily be differentiated with respect to $x'$ (tangential derivatives) since $\partial\_{x'}a\in S^m$. However the distribution $u$ is more difficult to differentiate wrt $x\_n$ since $\xi\_na\in S^{1+m}$; however calculating
$$
x\_n\partial\_{x\_n} u=\int\_{\mathbb R} x\_n\partial\_{x\_n}(e^{ix\_n\cdot \xi\_n}) a(x,\xi\_n) d\xi\_n
+\int\_{\mathbb R} e^{ix\_n\cdot \xi\_n} x\_n\partial\_{x\_n}(a)(x,\xi\_n) d\xi\_n.
$$
$$
=\int\_{\mathbb R} x\_n i\xi\_n e^{ix\_n\cdot \xi\_n} a(x,\xi\_n) d\xi\_n
+\int\_{\mathbb R} e^{ix\_n\cdot \xi\_n} x\_n\partial\_{x\_n}(a)(x,\xi\_n) d\xi\_n.
$$
$$
=\int\_{\mathbb R} \partial\_{\xi\_n}(e^{ix\_n\cdot \xi\_n} )\xi\_n a(x,\xi\_n) d\xi\_n
+\int\_{\mathbb R} e^{ix\_n\cdot \xi\_n} x\_n\partial\_{x\_n}(a)(x,\xi\_n) d\xi\_n.
$$
$$
=-\int\_{\mathbb R} e^{ix\_n\cdot \xi\_n} \underbrace{\partial\_{\xi\_n}(\xi\_n a(x,\xi\_n))}\_{\in S^m} d\xi\_n
+\int\_{\mathbb R} e^{ix\_n\cdot \xi\_n} \underbrace{x\_n\partial\_{x\_n}(a)(x,\xi\_n)}\_{\in S^m} d\xi\_n.
$$
We can prove in particular that the wave-front-set of $u$ is included in the conormal bundle of $Y$,
i.e.
$$
WF u\subset\{(x',0;0,\xi\_n)\}\_{x'\in \mathbb R^{n-1}, \xi\_n
\in \mathbb R^\*}$$
Your simple layer $u(x)= \omega(x')\otimes \delta(x\_n)$ with a smooth $\omega$ is indeed the prototypical conormal distribution since 
$$
u(x)= \int\_\mathbb R\omega(x') e^{2iπ x\_n\xi\_n}d\xi\_n.
$$
It is not difficult to generalize the previous discussion to the case involving $k$ codimensions.


