I'll frame my answer in terms of a comment by the original poster, because that comment asks the question most succinctly:
Please tell me any one inertial frame so that I can decide which frame is inertial and which is not.
I'll start my answer from a Newtonian perspective and assume that a (possibly unknowable) Newtonian inertial frame does exist. Every frame of reference invented by humankind inevitably will have two non-inertial aspects to it:
- The origin of the frame is accelerating with respect to an inertial frame, and
- The axes of the frame are rotating with respect to an inertial frame.
Some of those accelerations and rotations are known. For example, a frame fixed with respect to the rotating Earth is accelerating toward the Sun and Moon (and also toward other masses in the solar system, and outside it), and it rotates. These known accelerations and rotations are easily dealt with. One can either ignore them or take them into account, yielding a non-inertial frame with fictitious forces.
What about the unknowns? Every so-called inertial frame has later been found to be a rotating frame. Even our best inertial frames are but approximations. Astronomers have developed and then later discarded a number of supposedly non-rotating frames over the last 100 years. Some recent ones:
The FK4 frame, also known as the B1950.0 frame, also known as the M50 frame.
This frame used the fourth fundamental catalog of stars (FK4). There's a basic problem with using the "fixed stars" as the underlying mechanism for defining a non-rotating frame: The "fixed stars" aren't fixed. Astronomers try as best they can to deduce and then remove the effects of proper motion of the stars on the inferred frame of reference, but they can only go so far. The FK4 frame was later found to be rotating. That said, the FK4 frame got humanity to the Moon and the Voyager spacecraft to Jupiter, Saturn, and beyond. It also let astronomers peer ever deeper into the universe.
The FK5 frame, also known as the J2000 frame.
This frame used the fifth fundamental catalog of stars, an update to the FK4 catalog. This frame let astronomers peer even deeper into the universe.
The International Celestial Reference Frame, also known as the ICRF.
The improvements offered by the FK4 and FK5 frames let astronomers peer extremely deeply into the universe. Some of the very remote objects they found were very bright (after accounting for their remoteness). These pulsars provided a means to go beyond star catalogs. The very remoteness of the pulsars drastically reduces the problem of proper motion. After the fact, it has been found that the FK5 frame is rotating with respect to the ICRF by about 3 milliarcseconds per year. That's incredibly small, but it's also incredibly important to astronomers, who have been looking at the sub-milliarcsecond level for the last twenty years.
The second realization of the ICRF, also known as the ICRF2.
Improvements to the pulsar database kept by astronomers inevitably meant that the ICRF would itself be found to be rotating. And it was. The ICRF was updated in 2009 as the ICRF2. That is currently the best guess as to what constitutes a non-rotating frame. It too will eventually be found to be rotating.
The above explicitly assumes a Newtonian inertial frame exists. There is no such thing. While the inertial frames listed above got humanity to the Moon and let astronomers see back to the near the beginning of the universe, they are not universal. They can't be, thanks to general relativity.