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In some songs that I have listened, only one inversion could be used; when one tries to use other voicings, it will sound really odd. So, when it is more preferrable, for example, to use the second inversion than the root position and first inversion, and vice versa. I am specifically asking about the major chord inversions; I'll leave the other chords for other questions.

Example of a song I am refering to: the Hook part of IU - You and I. This songs uses Eb (V) --> Bb/D (I) --> Cm (ii) ... When I try to play Bb/D using Bb as the bass voice on the piano, it sounds really odd.

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Not having heard the song, I guess it'll be a bass figure, running down from Eb on the IV to D on the Bb, to C on the Cm. If it had been recorded with 3 roots, playing this may sound odd. It's partly what we get used to hearing.

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You can't really say it's "preferrable" in any particular circumstances, like you can't say it's preferrable to use a particular key – those are just musical decisions. You can compose songs without ever giving the bass a root note, though obviously it'll sound somewhat strange and lack familar resolution (but perhaps you want just that).

Where inversions tend to arise naturally is when you have a certain melody in the bass voice. In particular, when the bass goes through an entire diatonic (or even chromatic) scale then it's hardly practical to use each of those notes as a root of a major or minor chord – at any rate It'd make for pretty poor voicing. However using some thirds in the bass by going to the first inversion can make it much easier, as in your example: there's a simple E♭ - D - C bass melody (many will argue that melody always has precedence over harmony), which coincides with the simple deceptive cadence I - V - vi if you use the d as the bass note of the first inversion of B♭-major.

In popular music such diatonic runs are by far the most common application of chord inversions.

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Though we can lose sight of the fact when treating chords as separate entities (something guitarists can be prone to) harmony arose as the result of combining simultaneous melodic lines. And it is still true that a progression can be 'good' because of a strong melodic line within it, no matter what scale or mode they are drawn from.

Think of the common progression C, C7/Bb, F/A, Fm/Ab, C/G, G7, C. (That's the end of 'When the saints come marching in'). The bass line goes C, Bb, A, Ab, G, G, C. That's enough reason why it works. We don't need to worry about what mode Fm is 'borrowed' from or whether the tonal centre moves to F for a moment - that strong stepwise line is all we need.

Eb, D, C is a strong bass line. Eb, Bb, C is not 'wrong', but it has a very different flavour.

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