Path of a particle spiraling into a black hole

One of the aspects of general relativity that students find hard is that coordinates do not have a physical relevance. In Newtonian mechanics we are used to thinking of coordinates $(t,x,y,z)$ as meaning times we can measure with our clock and distances we can measure with our rulers. However in GR coordinates are just labels we apply to identify points in spacetime. We often try to choose coordinates that do correspond to measurements in an intuitive way but this isn't always convenient or indeed possible. For example in the Schwarzschild coordinates the $r$ is not a radial distance i.e. not what you would measure if you dangled a tape measure radially down towards the event horizon.

The one quantity that all observers will agree on is the proper time. In your example we would attach a stopwatch to the infalling object, and zero it when the object started its plummet. Then the elapsed proper time, usually denoted by $tau$, is just the time shown on the stopwatch. We describe the trajectory by using the proper time as a parameter, that is the position in spacetime is given by functions of the proper time:

$$ begin{align} t(tau) r(tau) theta(tau) phi(tau) end{align} $$

The falling object reaches the singularity in a finite (usually short!) proper time so for every value of the proper time $tau$ we can use the functions $t(tau)$, $r(tau)$, etc to assign a position $(t,r,theta,phi)$. In this sense the trajectory is well defined over the whole region outside and inside the event horizon.

The problem is that if we use the Schwarzschild coordinates we find the function $t(tau) to infty$ as the trajectory approaches the horizon. This doesn't mean there is anything strange about the trajectory there, an infalling observer wouldn't even notice when they crossed the horizon. It just means we have chosen coordinates that don't make sense at the horizon.

And the same applies inside the horizon. A lot of nonsense has been written about the fact that $t$ becomes spacelike and $r$ timelike inside the horizon. The trajectory inside the horizon is perfectly well defined and the problem is just that our choice of coordinates isn't useful. We get round this by switching to a different set of coordinates e.g. the simplest coordinates that remain reasonably intuitive are the Gullstrand-Painlevé coordinates, though for numerical work we'd probably use the (much less intuitive) Kruskal-Szekeres coordinates.

I've ranted on a bit here. The point I want you to take away is that the trajectory parameterised by the proper time is perfectly well defined everywhere. We may get weird results when we express it in our chosen coordinate system, but that's our fault for making a poor choice of coordinates.