I've taken apart the trap n cleaned it and also snaked the drain but still no water goes down! I noticed that the overflow is completely clogged..would this stop the drain from emptying? The overflow n sink r one complete Unit. ..how am I suppose to clear out the overflow when I can't get to it??
Assuming there are no other drain problems occurring (because you did not mention any) you most certainly have an obstruction of the branch drain line leading to that sink.
Option 1- run a good quality drain auger down the drain, through the trap, and down the branch line as far as it will go.
Option 2- take the trap back apart and run the cable directly down the drain line.
For either option, it is important that you use the correct technique and persevere until the length of the cable has gone down. It's not easy, be prepared to work. There are numerous instructional videos on-line, this one shows the correct technique but the person is using a cheap sub-standard auger.
I always cable from the top and run water until it backs up a bit, so I can tell when I have hit the clog (because the water suddenly drains). I highly recommend that you invest in a good professional quality auger, I did and it has saved me hundreds and hundreds of dollars over the years:
Always wipe off the cable with a rag as you remove it from the drain, and always spray it down with your favorite water displacing spray lube that comes in a blue and yellow can or it will rust in the drum.
The overflow in and of itself wouldn't do it... but it makes me suspect that the rest of the drain "system" is equally bad if the overflow is that bad. Use an old style plunger... cover the overflow with a wet towel, put water in the sink and pump the plunger up and down to dislodge whatever is in there. But DO NOT do this if you have already tried drain cleaner!