The question has been asked (e.g., here and here), but I would like to get a more definitive and mathematically formal answer.
The Earth rotates around its axis, around the Sun, and participates in larger scale motions as a part of the Solar system. Yet, we often can get by treating it as an inertial reference frame (e.g., when constructing furniture, cars and buildings). In some cases we do need to take account for the effect of its rotation – e.g., in weather prediction one takes account for the Coriolis force, but we still consider the Solar system as the inertial reference frame.
We do that because:
- accelerations that we deal with (notably $g$) are much greater than the accelerations due to the other motions that it is involved in?
- we can neglect the non-inertial forces because all the objects around experience the same accelerations due to these forces?
- something else?
I am looking for a mathematically motivated answer. I also suggest delineating between what is specific to Earth (accidental) and what would apply to all (or most) planets/stellar bodies.
Update
I took the liberty to summarize the opinions expressed so far in my own answer. Yet, there remains non-inertial effects not covered by free fall and the Earth's rotation – those related to the Earth's finite size and responsible for the tidal forces (more specific question is here). Thus, this question still needs a canonical answer.
Best Answer
If you need to throw a ball down a field, then corrections for the rotation amount to about 0.0001g acceleration. Small enough to be ignored for the 3 seconds the ball is in the air.
If you're shooting a projectile into another state, then that apparent acceleration is going to add up to many meters of deflection against the predictions when using an inertial frame for calculations.
Most human-scale experiences don't have sufficient distance or velocity for the error in assuming an inertial frame to become apparent. So the extra complexity of calculating the changes due to the earth's movement is unnecessary. The same way that for many benchtop experiments we might be able to ignore air resistance and the non-uniformity of the gravitational field.
There's no hard rule about this. You just choose the simplest model that is sufficient for your purpose. Want a pipe that takes water from the top of a 10m building to the ground? Probably can ignore earth rotation for how pressures in the pipe develop. Need to model how air circulates around the planet? You can't take that shortcut.
The accelerations due to earth motion are smaller than the accelerations that you notice in everyday life. It's not an accident, it's just that the earth rotates fairly slowly on the human scale.
Go get a merry-go-round, hook up a motor to it to make it rotate once every 24 hours. Stand on it and you won't be able to tell much is going on. Accelerations due to earth's orbit and due to galactic rotation take even longer and will be even smaller.
The Coriolis corrections are proportional to the rotation rate of the frame ($\omega$). One revolution a day just isn't very large.