Gerund

[Malcolm]

I remember that the following sentences are fine in formal register:

1. I heard Jamie's singing in the bath.

2. They remember the cast's rehearsing for days.

I was, however, told that the following types of sentences with pronouns like "someone" and "something" are incorrect; now, I do not remember the reasons. Could someone please tell me what the reasons are? I appreciate it:

3. The witness saw someone's running away.

4. We could smell something's burning.


Thank you very much

[Darryl]

In the first two sentences the apostrophe is clearly indicating possession. Jamie's singing. His singing.
However, in the other two sentences, the usual construction would be The witness saw someone running away and We could smell something burning. Because this is such a common expression, the addition of the apostrophe makes it sound unusual and creates confusion with a contraction of something is. On first reading of #3 and #4, you would not recognise the apostrophe as being one of possession.

[admin]

The apostrophe s is not needed in any of these - none of them need to be possessive constructions although, as Darryl says, they can be. It is quite an old fashioned form but would be "correct" for all for of your examples.

Much more current would be the ellipsis form (without the 's)

remember that the following sentences are fine in formal register:

1. I heard Jamie [who was] singing in the bath.

2. They remember the cast [who had been] rehearsing for days.

I was, however, told that the following types of sentences with pronouns like "someone" and "something" are incorrect; now, I do not remember the reasons. Could someone please tell me what the reasons are? I appreciate it:

3. The witness saw someone [who was] running away.

4. We could smell something [which was] burning.

[Bertha]

Gerunds and present participles can confuse users since both end with -ing. 

In your sentences, the first one could be either one.  If it's without the apostrophe, the -ing form is a present participle, which acts as modifier to the preceding noun (Jamie), similar to an adjective. So, "I heard Jamie" who is singing.  If an apostrophe appears, the -ing form is a gerund, which is a noun form, and it becomes the object instead of Jamie:  "I heard singing" As it happens the singing belongs to Jamie.  It is true that the latter seems more formal.

In the second sentence, it's the same really.  The choice of usage depends on the idea you want to provide and maybe it could be slightly more emphasis on the who or on the what.

When you have pronouns such as "someone," the tendency is to not use the possessive gerund.