Trajectories in Standard quantum mechanics
Standard quantum mechanics does not involve trajectories for individual particles. For simplicity, let's focus on a one-particle wave function (the argument also applies to the case of several particles).
A trajectory of a single would be given by an assignement
$$ X: \mathbb T \to \mathbb R^3\,,\enspace t \mapsto X_t\,,$$
where $\mathbb T$ is the time.
Standard quantum mechanics is a probabilistic theory. So the position of the particle $X_t$ is described by a random variable. According to the Born rule, the probability distribution of $X_t$ is given by
$$\mathbb P_{\psi_t}( X_t = x) = |\psi_t(x)|^2 dx\,.$$
However, the standard formulation fails to specifiy a joint probability for random variables $\{X_t\}_{t\in \mathbb T}$. If given such a joint probability measure, one could calculate the probability for the particle's trajectory to be $\gamma: \mathbb T \to \mathbb R^3$:
$$
\mathbb P( X_\bullet = \gamma_\bullet) = \mathbb P( \forall t: X_t = \gamma_t)\,.
$$
Note, that taking the product measure treating the $X_t$ as independent, i.e.,
$$
\mathbb P( X_\bullet = \gamma_\bullet) = \prod_{t \in \mathbb T} |\psi_t(\gamma_t)|^2 dx\,, \tag{1}
$$
does not yield physical trajectories, as the particle would randomly jump between different places. This gets even worse if the many particle case is considered. E.g., the ensemble of particles making up Schrödinger's cat would randomly jump between dead cat configurations and live cat configurations. However, this would not affect the histories we remember as our brain would constantly jump between a brain remembering a cat that died and a brain that remembers a cat that survived.
Trajectories in the phase space formulation of quantum mechanics
In the phase space formulation of quantum mechanics, instead of a wave-function or density matrix the quantum state is described by a quasi probability distribution $W(x,p)$. The time evolution is given by
$$
\frac{\partial W}{\partial t} = - \{\{ W, H\}\} = - \{ W, H\} + \mathcal O(\hbar^2)\,, \tag{2}
$$
where the Moyal bracket $\{\{\bullet, \bullet\}\}$ is a deformation of the classical Poisson bracket. For $\hbar \to 0$ the equation would reduce to the classical Liouville equation. The difference for non-vanishing $\hbar$ is that $(2)$ is not a flow, because the density of points is not conserved. That is, the evolution of the signed measure $\mu_t(A) = \int_A W_t(x,p) dx dp$ is not given by a flow $\Phi_t : \mathbb R^{2n} \to \mathbb R^{2n}$ such that
$$\mu_t(A) = \Phi_t{}_* \mu_0(A) = \mu_0(\Phi_t{}^{-1}(A))\,.$$
this flow would be needed to define the trajectories. In classical mechanics, the trajectories for initial configuration $(x_0, p_0)$ would be defined by $(x_t, p_t) := \Phi_t(x_0, p_0)$, where $\Phi_t$ is the flow corresponding to the classical phase space measure $\rho$.
Trajectories in Bohmian mechanics
Bohmian mechanics is a proposal for a rational completion of quantum mechanics. According to Bohmian mechanics, the wave function always evolves according to the Schrödinger equation
$$
\mathrm i \hbar \frac {\partial \psi} {\partial t} = \hat H \psi\,
$$
and the particle positions $\vec Q_i(t)$ move according to the law
$$
\frac {\mathrm{d} \vec Q_i(t)} {\mathrm{d}t} = \frac {\hbar} {m_i} \operatorname {Im} \frac {\vec \nabla_i \psi} {\psi}\,.
$$
As a consequence of the continuity equation, if the initial configurations $\vec Q_i(0)$ are distributed according to Born's rule
$$ |\psi_0(x_1, \dots, x_n)|^2$$
the configurations will be distributed according to Born's rule
$$ |\psi_t(x_1, \dots, x_n)|^2$$
for any later time.
One way to view this is that Bohmian mechanics provides the joint probability measure that was missing in standard quantum mechanics. The difference between the Bohmian mechanics and the measure defined by $(1)$ is that Bohmian mechanics yields physically reasonable trajectories. In particular, according to Bohmian mechanics, we can trust our memories since Bohmian trajectories do not randomly jump between distinct branches of the wave function. E.g., if my brain is in the state remembering that I observed a cat alive for the last ten minutes, the most likely the Bohmian particles that constitute the cat were in configurations corresponding to a cat alive for the last ten minutes (There are of course caveats due to neuro-science/biology since humans can have false memories).
I hope this gives an overview of the status of trajectories in quantum mechanics.
