-ing participle clauses

 Which is correct? 'I could hear the dog barking' or 'I could hear the dog's barking'?

I could hear the dog's barking is grammatically correct. The dog's barking uses a possessive 's and means the barking of the dog.

However, you have used the verb hear which gives the idea that you could hear something that was happening at the time you are speaking about. So it would be more appropriate to use the participle clause barking to give more information about the dog. The preferred phrase would be I could hear the dog barking.

So let's look more closely at -ing participle clauses.

We use –ing participle clauses to give more information about what was happening at a particular time.

•  Did you speak to the man wearing the grey jacket? (the man who was wearing a grey jacket)

•  The lady working behind the counter is very busy. (the lady who is working behind the counter)

•  I could hear the dog barking. (the dog which was barking)

We can also use –ing participle clauses to talk about something that happens all the time.  

•  At the end of the road there is a path leading to the station. (a path which leads to the station)

•  The agent has a website giving details of all the properties for sale. (a website which gives details of all the properties for sale)

With the verbs see, hear, watch, and listen to, the meaning can change depending on whether an -ing participle is used. Compare these sentences:

•  Stella saw the children cross the road. •  Stella saw the children crossing the road.

In the first one - Stella saw the children cross the road - the use of cross in the simple form implies that Stella saw the complete action – she saw the children cross the road from start to finish. But in the second one - Stella saw the children crossing the road – the use of the participle suggests that Stella saw the children while they were crossing the road – she didn't see the complete action. So in your example sentence Arun, I could hear the dog barking suggests that the dog was already barking when you heard it – and probably continued barking even when you had left.