The sociolinguists have demonstrated that most native speakers of English fluctuate between two forms of the -ing ending: the ‘high’ variant ɪŋ, with a velar nasal, and the ‘low’ variant ɪn, ən, with an alveolar nasal.
The difference is stylistic, with the H variant being used in formal situations and the L in informal/colloquial situations. Just where the line is drawn between the two possibilities varies, depending on social class and other factors.
Using the L variant is popularly known as “dropping one’s g’s”, although in surface phonetics it is a matter of place of articulation rather than of omitting something. It is of course shown in writing by the use of an apostrophe, putting -in’ in place of -ing. (The tattoo in my picture omits the apostrophe.)
The L variant also has the subvariant of a syllabic n̩, used particularly after t (→ ʔ) and d. Both ɪn and n̩ are obviously also subject to possible dealveolar assimilation, producing forms with m and (!) ŋ.
So for running we can have H ˈrʌnɪŋ or L ˈrʌnɪn, ˈrʌnən. For putting we have H ˈpʊtɪŋ and L ˈpʊtɪn, ˈpʊtn̩, ˈpʊʔn̩. Because of possible assimilation, if we are faced with ˈrʌnɪŋ ˈkwɪkli or ˈteɪkŋ̩ ˈpleɪs we cannot tell which of H and L is involved.
However at the top of the social scale there is a smallish group of speakers who use H (velar) under virtually all circumstances. At the bottom there is a rather larger group who virtually always use L (alveolar).
Personally I have to admit membership of the first group. I don’t believe I ever use the L variant except for jocular or other special effect.
The lyrics of popular music often mandate the L variant. I find it difficult to remember to do this as required when singing.
I remember when BBC English assigned me a producer (many years ago) who used the L variant extremely frequently. I was shocked at my own gut reaction, which was that he must be ignorant and uneducated to the extent that I found it difficult to take his opinions seriously.
Because it is so socially sensitive, this variable also generates hypercorrections. In Accents of English (p. 263) I mentioned as examples of this a braz[ɪŋ] (brazen) hussy and Badmi[ŋ]ton. The other day I heard a nice one from a railway station announcer: Harpenden pronounced as ˈhɑːpɪŋdən.