Much more than Iwasawa's original theorem is known by now. First of all, the $p$-primary part of the class group stays trivial in the field $K_n=\mathbb{Q}(\mu_{p^{n+1}})$ unless it is already non-trivial for $K_0=\mathbb{Q}(\mu_p)$. See Proposition 13.22 in Washington's "Introduction to cyclotomic fields". Therefore the only primes of interest are the irregular primes and the first one is $p=37$. Ribet's results showing the converse to Herbrand's theorem lets you even determine the structure of the class group of $K_0$ with its action by the Galois group of $K_0/\mathbb{Q}$ in terms of Bernoulli numbers. It is known that $\mu=0$. Furthermore, the main conjecture is known, which means that the constant $\lambda$, and in fact most things about the limit class group as a $\Lambda$-module, can be determined from a $p$-adic zeta-function that can be calculated either using Stickelberger elements (so Bernoulli numbers) or modular forms. The constant $\nu$ and the constant $n_0$ such that Iwasawa's theorem is valid for $n\geq n_0$ can often be determined, too.
A lot of these things are in Washington's book. Lang's "Cyclotomic Field" is another standard reference.
I know too little about explicit calculations of these invariants. But I would expect that it becomes harder as the irregularity of the prime increases.
For the special case of $p=37$, I once did the explicit calculation for a lecture series in Heidelberg. The Bernoulli number $B_{32}$ is divisible by $37$. Accordingly, we expect a non-trivial $\omega^5$ part in the $p$-primary part of the class group of $\mathbb{Q}(\mu_{37})$ where $\omega$ is the Teichmüller character. Indeed the approximation to the $37$-adic $L$-function for this character is
\begin{multline*}
14\cdot 37 + 33\cdot 37^2 + 13\cdot 37^3 + \mathbf{O}(37^4) + \bigl(16 + 6\cdot 37 + 32\cdot 37^2 + \mathbf{O}(37^3)\bigr)\cdot T \\
+ \bigl(29 + 9\cdot 37 + 13\cdot 37^2 + \mathbf{O}(37^3)\bigr)\cdot T^2 + \mathbf{O}(T^3).
\end{multline*}
This is not a unit as $-B_{1,\omega^{-5}}$ is divisible by $37$. From the fact that the second coefficient is a unit, we conclude that the $p$-adic $L$-function is a unit times a linear factor. Hence the limit class group $X$ is a free $\mathbb{Z}_{37}$-module of rank $1$ and hence the $p$-primary part of the class group of $K_n$ is of order $p^{n+1}$ for all $n\geq 0$, i.e., $c(n)=n$ in your notations.
The fact which underlies the proof of Ribet's theorem is that the Eisenstein series
\begin{align*}
G &= -\frac{B_{32}}{2\cdot 32} + \sum_{n\geq 1} \sum_{d\mid n} d^{31} q^n \\
&={\scriptstyle \frac{7709321041217}{32640} + q + 2147483649\,q^2 + 617673396283948\,q^3 + 4611686020574871553\,q^4
%+ 4656612873077392578126\,q^5
+\cdots}
\end{align*}
of weight $32$ is congruent modulo one of the primes above $37$ in $\mathbb{Q}(\mu_{12})$ to the cuspform
$$
f = q + \zeta_{12} \,q^2 + \bigl(-\zeta_{12}^3 +\zeta_{12}^2 -\zeta_{12}\bigr)\,q^3 -\zeta_{12}^2\, q^4 + \bigl(2\,\zeta_{12}^3 +\zeta_{12}^2-2\,\zeta_{12}^2 -2\bigr)\, q^5 + \cdots
$$
of weight $2$ for the group $\Gamma_1(37)$ and character $\omega^{30}$.
