The dirty price is what settles. It is the number on the trade confirmation and the number your cash balance falls by. It jumps down on every coupon date as accrued resets to zero, which is exactly why nobody quotes it.
When you compare a bond's return to anything else, the dirty price is the honest cost basis. A model that discounts future cash flows and solves for yield-to-maturity is solving against the dirty price, not the clean one.
Example: clean price 98.50, accrued 1.10, face $1,000,000. Dirty price is 99.60, so the invoice is $996,000. The screen still shows 98.50.
Related: clean-price, accrued-interest, yield-to-maturity, bond