I think that the most famous and beautiful trajectory of the $3x+1$ problem is without doubt that starting from $n=27$ and having a maximum at $9232$.
The thing that I find very beautiful is that:
$$19\cdot 3^3=513\equiv 1\pmod {2^k}$$
And
$$\frac{9232}{2}=19\cdot 3^5-1$$
Is it chance or there is a Deep connection with the prime $19$ and the order of $19\cdot 3^s\pmod {2^k}$?
Are there other trajectories with similar features?
That's pretty common in the Collatz trajectories. Take any odd number and write it as $a\cdot2^n-1$, it will always climb up to $a\cdot 3^n-1$ in exactly $n$ steps of the function $f(x)=\frac{3x+1}{2}$. From there it will be divisible by $2^k$ with $k>0$. So just choose $a$ prime and you will find plenty of them with various value of $k$. $$a\cdot3^n\equiv 1\bmod(2^k)$$ That's called a "1-cycle"