The Iraqi air defense system, including it's aircraft were distroyed in the first 48 hours. A few Iraqi planes escaped to Iran. They shot down nothing.
That is not entirely accurate... The Iraqi air force did shoot down and/or severely damage a number of coalition aircraft. The Iraqis lost 23-44 aircraft in air-air combat varying on report. The coalition lost 4-5 destroyed and 4 damaged. So the coalition enjoyed a K/D of between 5:1 and 10:1.
On the first night of the war, two F/A-18's from the carrier USS Saratoga were flying outside of Baghdad when two Iraqi MiG-25PDs interceptors from the 96th Squadron engaged them. In the beyond-visual-range (BVR) kill, one of the Iraqi MiGs piloted by lieutenant colonel Zuhair Dawood, fired an R-40 missile. The missile impacted Scott Speicher's jet head on when he was travelling Mach 0.92. The impact sent the aircraft spiraling downwards and most people believe Speicher died on the impact of the missile.
An Iraqi MiG-23 fired a R-24T missile at a F-111 on a bombing run and scored a hit, although the bomber made it safely back to base. Another similar incident occurred with the same Iraqi interceptor several minutes later, this F-111 also made it back to base despite the severe damage to the aircraft.
An Iraqi MiG-29 struck an F-111 aircraft with a R60 missile, though the sturdy F-111 stayed airworthy. Several minutes later the same pilot fired a R27 missile at a B-52G on a bombing run, severely damaging it.
It has been claimed by some sources that a Tornado (ZA467) crewed by Squadron Leader Gary Lennox and Squadron Leader Adrian Weeks was shot down on 19 January by a R-60MK missile fired from an Iraqi MiG-29 piloted by Jameel Sayhood, however this aircraft is officially recorded as having crashed on 22 January on a mission to Ar Rutbah.
In what was the last aerial victory for the Iraqi Air Force before Operation Iraqi Freedom, an Iraqi MiG-25 destroyed an American UAV RQ-1 Predator after the drone opened fire on the Iraqi aircraft with a Stinger missile.