If it is a PVC compression fitting, put a bucket under the p-trap and take the trap off.
Odds are good that you will find one of two things. Either the threads were cross threaded, which will affect the pressure seal around the pipe, or the plastic o-ring was caught in the threads at one point in time and was mangled. A mangled o-ring won't provide a satisfactory seal.
Either way you will know which part to replace, and these connections (provided it is compression fitting PVC) can be replaced easily at your local hardware store.
If it seems that everything is fine, look for the following items:
Is the pipe being unnaturally flexed or bent due to particular pipes being too long or short? If so the solution is shorter / longer pipes, but make sure you don't dramatically change the geometry of your setup.
Are the locking nuts really tightened appropriately? Sometimes a good "extra twist" solves a leak in the best possible way. Remember, if you "break" an obviously replaceable section of pipe, you're out a few dollars in plastic, so don't be shy.
The joint between the end of the p-trap and the pipe from the wall is supposed to be constantly under water, so it is fairly common for those to leak. Normally they are fitted and tested for leaks. The workman will not leave with one leaking. Since it is under the sink, and people often store items under their sink, eventually the pipe gets bumped. That's probably the source of your new dripping; however, it is highly unlikely that anyone will remember bumping the pipe, even if they did it.
The photos posted above were really helpful, since many people may not know what connectors and adapters are available -- as I didn't before replacing two vanities. Therefore, I thought I'd post photos of what I did and the components I used.
The first thing to do is come out of the wall with a wall tube (otherwise known as a quarter-bend wall tube). Cut the long side if necessary, so it slides into the drain tube in the wall (make sure you have at least several inches in the wall drain).
![Quarter-bend wall tube](https://i.stack.imgur.com/OFPDk.jpg)
The flange typically connects to the short leg of the J-trap.
![First vanity - before](https://i.stack.imgur.com/YqViq.jpg)
There are two types of J-traps: one with a threaded slip-joint fitting on each end and a "repair" J-trap, with a slip-joint on the longer leg and a compression flange on the other. Use whichever one is suitable for your existing piping.
![J-trap](https://i.stack.imgur.com/7GB5E.jpg)
![Repair J-trap](https://i.stack.imgur.com/wkv1B.jpg)
The connectors you use depend on two factors: how far is the tailpiece horizontally from the drain connection, and how far above or below it?
In the above situation, the drain connection is several inches to the side of the tailpiece and slightly below it. To close up horizontal distance, you can use another wall tube cut to the appropriate length, connected to the J-trap. The other end connects to a 90-degree elbow to make the turn and cover the vertical distance. Use a reducing washer to adapt a 1 1/4" tailpiece to 1 1/2" drain piping. (All the components pictured here are from Home Depot; other stores have similar items).
![90-degree elbow](https://i.stack.imgur.com/QlKRE.jpg)
![First vanity - after](https://i.stack.imgur.com/pvnuL.jpg)
My other vanity has greater horizontal distance and less vertical. Therefore, I connected the longer leg of the J-trap to the wall tube, to get more displacement down, so the connectors would have room to cover the horizontal distance on the way back up. In this case, the 90-degree elbow didn't quite make it to the bottom of the tailpiece, so I had to add a coupling, attached to the elbow with a scrap piece of the wall tube I had cut off.
![Second vanity](https://i.stack.imgur.com/BmPkj.jpg)
If, for some reason, you need even more vertical distance, you can use an extension tube, which has a threaded slip-joint fitting on one end, and connect it between the wall tube and the J-trap. (I tried this first, then realized I didn't need all that vertical distance and could eliminate the extension tube.)
![Second vanity with unneeded extension tube](https://i.stack.imgur.com/ZbRiT.jpg)
If you find that it leaks after you tighten everything up, and you have water on top of the reducing washer on the tailpiece, it means your drain assembly isn't watertight against the sink. You may need to tighten it (carefully, so as not to crack the sink) or remove it and apply more silicone or plumber's putty, along with Teflon tape on the threads.
Best Answer
I believe Oregon uses the UPC as the basis for their state/local code. Under the UPC S traps are illegal, and that's what you have there. S traps are illegal because they can cause the trap seal (the water that prevents the sewer gasses from entering your house) to siphon out.