-ing vs -in' ending

Solution 1:

The 'g' in -ing is never pronounced. What is pronounced is the velar nasal consonant represented in IPA as [ŋ]. In some dialects, this is replaced by the alveolar nasal consonant represented in IPA as [n]. This is the phonetics that the -in' ending represents.

The difference between [ŋg] and just [ŋ] can be heard in the difference between the words finger and singer.

You should never use a [g] in meeting. Use [ŋ] (which is usually represented in English spelling as 'ng') and not [n], [ŋg], or [ng].

The local dialect in several regions of the U.S., and apparently in parts of Britain as well, uses [ɪn] rather than [ɪŋ] for the suffix -ing. This is sometimes spelled -in'. The people speaking these dialects can pronounce the consonant [ŋ] just fine; for example, singin' would be pronounced [sɪŋɪn]. For more information on this, see this dialect blog posting.

Solution 2:

Using an apostrophe in place of a "g" is an informal colloquiallism. It is usually found inside a quotation, suggesting the speaker did not use much care when enunciating. Specifically, it's used to indicate the verb was spoken such that the final "g" was omitted: "We were walkin' to the store, not botherin' nobody, when, all of a sudden, out o' nowhere, this guy starts a-hollerin' at us for no good reason!"

(It's similar to using o' in place of "of " - as found in the preceding example).

Sometimes it's also used in song titles, when the singer doesn't carefully enunciate the final "g" in an "-ing" verb (a la "Takin' Care of Business" and "The Times They Are a-Changin'").

Solution 3:

The velar nasal ng lost the final g in English around the end of the 16th century. The phenomenon is called ng-coalescence. From Wikipedia:

As a result of Ng-coalescence, Middle English [sɪŋɡ] sing came to be pronounced [sɪŋ]. As well as in word-final position, Ng-coalescence was applied also in cases where a verb ending in -ng was followed by a vowel-initial suffix, so *si***ng***ing* and *si***ng***er* also underwent the change.
Otherwise, word-internal -ng- did not undergo coalescence and the pronunciation [ŋg] was retained, as in finger and angle...
Some accents, however, do not show the full effects of Ng-coalescence...

However, in Romanian (a language I know) the g at the end is pronounced such as in the words: luɲg. Not all ng are ɲg such as kreangə , ungʲ . 1
Notes: 1) ŋ may appear or not in various languages. In some languages it cannot appear in the front position (e.g. English) or in the final position. In some languages it can appear in all positions.
See World Atlas of Languages.
2) ŋ can be seen as a phoneme or as an allophone of n before g or k
3) k remains next to ŋ in nk: tæɲk (English), taɲk, , mɨɲkare (Romanian)

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