"stinking air" is correct and usable in written English.
It is usually used figuratively to describe a bad atmosphere or feeling, and is often used as an intensifier. For example, "The stinking air of doubt and fear filled the room, leaving the team feeling completely demoralized."
In the mine, Sonny descends into deadly dust, deadlier methane gas, stinking air and falling rocks, crumbling roofs and "gob" of waste coal and water.
After the war (1867), Joseph Lister invented antiseptics (using diluted carbolic acid) yet in 1873 doctors commonly explained infections as damage from stinking air (miasma) that we know was odor of gangrene, dead flesh.
On its front, which looks out on an old plaza — Washington Plaza — that has been slowly and inexorably transformed into a noisy, greasy, stinking open-air municipal bus terminal, is a terra-cotta portico supported by four polished-granite columns, and on its south side, which looks out on the Williamsburg Bridge, is a terra-cotta gate, also supported by four polished-granite columns.
On its front, which looks out on an old plaza Washington Plaza that has been slowly and inexorably transformed into a noisy, greasy, stinking open-air municipal bus terminal, is a terra-cotta portico supported by four polished-granite columns, and on its south side, which looks out on the Williamsburg Bridge, is a terra-cotta gate, also supported by four polished-granite columns.
Kate Hanni, an advocate from Napa, Calif., for the rights of airline passengers, had tried for years to persuade the government to address a series of incidents in which flight delays left passengers trapped for hours on planes that had already left the gate, often in cabins with stinking toilets, weak air-conditioning and no food.
"The air is stinking, it's horrible, it's dirty".
But still that air fuel is stinking heavy.
We write of creaking bullock carts, curled moustaches, stinking latrines, sallow-cheeked farmers, smoky air, sweltering megalopolises and aching villages.
The air was stagnant and stinking.
There were flecks of soot in the air which, like the river, was "stinking".
If we don't air them out every once in a while, they start stinking everything up.
