All reference frame are inertial? Where is the flaw in reasoning? This is my first post here. I have a line of thoughts which might be wrong but I couldn't find the error. 
Suppose we have a point mass subjected to an arbitrary force $F$. In the lab reference frame we see it accelerating. In the reference frame where the mass is at rest we see the mass at rest. In this frame we don't see any force so the first law of dynamics is respected. We must conclude that this is an inertial reference frame and therefore every time I put myself in such a reference frame I am always in an inertial one.
This is clearly wrong since the reference frame is accelerating with respect to the lab reference frame and therefore it can't be inertial.
Where's the flaw?
 A: The flaw is that you've failed to do an experiment which will tell you whether the frame is inertial or not.  If you do such an experiment -- for instance take a test mass, initially at rest with respect to the frame, release it, and see if it remains at rest -- you will immediately discover that the frame is not inertial.
A: The flaw is your assumption that 

In this [accelerating] frame we don't see any force so the first law of dynamics is respected.

In the accelerating reference frame you do see evidence of a force, even though you don't see the effect you are expecting (acceleration of the object).
Like an observer standing on the surface of the Earth, acted on by the Earth's gravitational field, the object experiences the push of a contact force - eg from the ground. Or if the object is being rotated in a circle on the end of a string, it experiences a pull from the string.  If the object is elastic there will be some deformation (contraction, elongation, change of shape), which indicates that a force is acting. 
As tbf and CuriousOne point out, you could also actively perform an experiment to test if you are in an inertial frame, rather than passively observe.  EG release a test mass, or weigh it. 
