A bomb also went off at a famous monument in a Baghdad square honouring the 8th-century founder of Baghdad to whom Saddam had often compared himself.
The blast, which toppled the bust of Abu Jafar al-Mansur but caused no injuries, appeared to be a jab at Saddam.
Shortly before Saddam's trial began in Baghdad's highly secured Green Zone, attackers shot and killed Hakim Mirza, one of several municipal directors of the capital, and his driver in the southern neighbourhood of Dora, said police Major Falah al-Muhammadawi.
Elsewhere, assailants shot and killed Muhsin Chitheer in front of his home in the southwestern section of Baghdad known as al-I'alam, said police 1st Lieutenant Mutaz Salah al-Din.
Chitheer had been a lieutenant-colonel in the Iraqi army that US forces disbanded after invading Iraq in 2003.
The bombing of the famous monument honouring al-Mansur knocked his bust off the top of a 10-metre-tall triangular monument, said police Captain Qasim Husain.
The attack occurred at 1.30am in a northwestern area named after al-Mansur, a religious leader of the Muslim empire, who built Baghdad on the banks of the Tigris River in 762 AD.
A blast knocked down the bust of
During his rule from 1979 to 2003, Saddam often tried to compare himself and his accomplishments to those of Baghdad's founder.
Husain said it was not immediately known who had launched the attack or what motivated it.
In other violence in Iraq, a roadside bomb hit a US army patrol late on Tuesday night, killing one soldier and wounding two near Iskandariya, 50km south of Baghdad, the military said.
The attack raised to at least 1981 the number of US military members who have died since the beginning of the Iraq war, according to an Associated Press count.
A US and British soldier were
killed in separate roadside blasts
A British soldier also was killed by a roadside bomb late on Tuesday night in the southern region of Basra where most British forces are based, Britain's Ministry of Defence said.
Meanwhile, Iraqis continued to await the outcome of last weekend's constitutional referendum, as the slower than expected vote counting continued.
Questions about the integrity of the vote and physical barriers to getting marked ballots to the capital mean final results from the landmark referendum would not be announced until Friday at the earliest, officials said.
Iraqis voted in a referendum on
the charter on 15 October
Questions raised over the returns have prompted an audit.
The audit Iraq's Electoral Commission is conducting will examine results that show an unusually high number of "yes" votes - apparently including in two crucial provinces that could determine the outcome of the vote, Ninevah and Diyala.
The commission and United Nations officials supervising the counting have made no mention of fraud and have cautioned that the unexpected votes are not necessarily incorrect.
But Sunni Arab leaders who oppose the charter have claimed the vote was fixed in Ninevah and Diyala and elsewhere to swing them to a "yes" after initial results reported by provincial officials indicated the constitution had passed.
Both provinces are believed to have slight Sunni Arab majorities that probably voted "no" in large numbers, along with significant Shia and Kurdish communities that were expected to largely cast "yes" ballots.
Initial results from election officials in Ninevah and Diyala indicated that about 70% of voters supported the charter and 20% rejected it.
Sunni Arab opponents needed to win over Diyala or Ninevah to veto the constitution.
Sunni Arabs had to get a two-thirds "no" vote in any three of Iraq's 18 provinces to defeat the charter, and they appeared to have got it in al-Anbar and Salah al-Din, both heavily populated by Sunni Arabs.
The minority group fears the constitution will divide Iraq into three districts: Oil-rich Kurdish and Shia mini-states in the north and south, and a mostly Sunni Arab region in western and central Iraq that would include a weak government in Baghdad.