Newton stated his first law as "Every object perseveres in its state of rest, or of uniform motion in a right line, except insofar as it is compelled to change that state by forces impressed thereon." We can rephrase Newton's first law a little bit as 'An inertial reference frame is a frame where inertial objects continue in a state of rest or in a state of constant velocity in a straight line.' Here, an inertial object is an object that feels no proper acceleration, so an object in free fall can be an inertial object, even though it may appear to be accelerating from the point of view of a non-inertial observer. The surface of the Earth does not qualify as an inertial reference frame by this definition because if you release a ball from your hand, it obviously does not continue in its state of rest and accelerates towards the ground.
We can determine an observer is not in an inertial reference frame, if they experience proper acceleration. This is not really a separate condition to using Newton's first law to define an inertial reference frame, because the lack of proper force on an inertial observer is already implied by the first law.
Onboard an orbiting space station, an observer does not measure any proper force acting on themselves and they can consider themselves to be locally in an inertial reference frame. In Newtonian terms, the centrifugal force of the orbiting space station is said to balance the gravitational force that provides the inward centripetal force. The fact that an observer in orbit cannot measure any proper force acting on themselves, is because centrifugal force and the force of gravity are both fictitious forces and cannot be measured by an accelerometer attached to an inertial observer.
- Global but not local inertial frame
It might be tempting to think that an observer in a rocket with proper acceleration in the hypothetical flat space of Minkowski spacetime might satisfy this condition. Certainly, the local reference frame of the observer is not inertial, but neither is the global flat space. Any test particles fired out into the flat space will not travel in straight lines from the observer's point of view, and so, the reference frame cannot be described as inertial on a global scale. As MakiseKurisu mentions in his answer, a global reference frame cannot exclude the local part, so no scenario can satisfy this pair of requirements.
- Local but not global inertial frame
A free-falling observer falling towards a gravitational body can be described as being in an approximately inertial reference frame, locally. The local reference frame, in this case, is not an exact inertial reference frame, but the smaller the space and time under consideration, the closer the local volume approximates an inertial reference frame.
If an observer is falling towards an spherical gravitational body, they can arrange a spherical shell of particles around themselves to determine the local curvature of space. It is a curious fact that in curved spacetime this spherical shell will contract horizontally and elongate vertically over time while maintaining constant volume. If the sphere is small enough that the distortion is difficult to determine, then the volume of the shell can be considered as the volume within which the reference frame can be considered to be inertial for practical purposes.
- A frame that is neither locally or globally inertial
Examples of locally non-inertial reference frames are the reference frames of an observer on an accelerating rocket, an observer on a carousel and an observer on the surface of the Earth. If a reference frame is not inertial at the local level then it is automatically not inertial at the global level. However, it should be noted that a reference frame that is not inertial can be approximately inertial over a small enough (possibly infinitesimal) interval of time and space. This is what is behind the concept of an instantaneous co-moving inertial reference frame that allows us to analyse an accelerating particle. It is also closely related to the "clock hypothesis", which allows us to determine the instantaneous time dilation of an accelerating clock by considering only its instantaneous velocity and completely ignoring its acceleration.
- A frame that is both locally and globally inertial
A frame that is globally inertial is automatically locally inertial (but not vice versa). In reality, there is no globally inertial reference frame in our universe that contains gravitational matter and curved space. The only scenario that qualifies is an inertial observer in the hypothetical flat space of Minkowski spacetime, which assumes an empty universe with no gravitational matter anywhere.
A frame attached to an observer free falling towards an infinite plane with a uniform gravitational field, would be very close to a frame that is both locally and globally inertial. The lack of tidal forces would make the reference frame appear inertial over considerable distances, but it debatable if this scenario can be considered to be a global inertial reference frame in the most general sense. Certainly the reference frame has a limited time span because it will inevitably hit the gravitational surface.
Although it does not fit neatly into one of the above categories, the case of the set of Rindler observers is an interesting example. If they are all moving in the same direction, they all appear to be stationary with respect to each other. However, if they release test particles at different heights, they will see the inertial test particles move away from each other over time indicating they are not in an inertial reference frame. They could of course arrive at the same conclusion, by simply measuring their own proper accelerations using personal accelerometers and noting that they do not have zero proper acceleration.
