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I understand that a general swap has 4 curves attached to it: the flat forecast curve associated with the fixed leg, the forecast curve associated with the floating leg, the fixed leg discount curve and the floating leg discount curve.
I also understand that $DV01 = \dfrac{\partial V_{swap}}{\partial y}$, where $y$ is the yield.
What I don't understand is which curve(s) is the $y$? Which curve are we shifting by $1bp$ to calculate the $DV01$?
Let's step back and look at the reason for making a DV01 calculation first before answering the question; The reason for making a DV01 calculation is to quantify what market movements has impact on the valuation of the trade.
Since the 'flat' forecast curve won't be affected by market movements the answer is (using pre-2008 methodology): The floating forecast curve.
After 2008 the discount curves became more important regarding the valuation as the previous standard to discount on the floating forecast curve (aka. IBOR-curve) was replaced by discounting on OIS (Overnight Index Swap)-curves or discount curves based on the collateral posted by the trade counter party.
In such a case DV01 would be calculated for the forecast curve, and for the discounting curve (which should be the same for both legs of the swap as long as both legs are in the same currency), resulting in two DV01 measurements.
The short answer is that there's no consensus. A popular method is to shock each input instrument by 1 bp (i.e., change the futures rate by 1bp, the swap rates by 1bp, OIS rates by 1bp, etc.), rebuild the curve, and then reprice the instrument of interest to obtain its curve sensitivity. This of course is not quite a "parallel" shift of any curve (e.g., a 1bp change in futures rate won't correspond to a 1bp change in either the zero curve or the par curve), but it's close enough and it does make hedging easier (after all, you're shocking tradable instruments).
Agree with Helin. For short term risk management a trader would be usually looking at delta to the forecast curve (i.e. swaps curve and government curve for a swaps/options trader), although he/she would also have delta risk to the OIS curve and also to curves of other currencies now that multi-currency collateralisation is quite common. Those other deltas tend to build up more slowly and over time (and therefore account for a smaller % of daily PnL), and so are managed/hedged on a slightly longer time scale.
