It was cancelled because KM Hasan, the man due to head the government, was ill, a presidential spokesman said.
The opposition says Mr Hasan, a retired chief justice, supports the ruling Bangladesh National Party (BNP) and cannot be trusted to run the interim administration tasked with organising the polls.
The opposition Awami League has threatened to paralyse the country if Mr Hasan takes office, with demonstrations and blockades of roads, railways and the country's main port.
Opposition MP Mirza Azam told supporters: "KM Hasan takes the responsibility as the chief of a caretaker government, right from that moment we will bring the country to a state of collapse."
Bombs
At least 10 people were reported wounded during Saturday's unrest.
According to Bangladesh TV, more than 14 bombs exploded in Dhaka alone.
Under Bangladesh's unique system, when an administration comes to the end of its term, it hands over to an unelected interim government which has 90 days to organise elections.
But transitions are often bitter - and currently the opposition is demanding a say in who will be the chief adviser of the caretaker PM.
Mrs Zia's mandate expired at midnight on Friday. She appealed for calm as a new wave of violence hit towns and cities across the country.
She promised the poll would be free and fair, and said the constitution would "safeguard a peace-loving and responsible nation".
It was a matter of "great regret" that talks with the opposition aimed at defusing the current crisis had not produced results, she added.
Not on speaking terms
The BBC's Roland Buerk in Dhaka says Bangladeshi politics is especially bitter because of the personal rivalry between Sheikh Hasina of the Awami League and Mrs Zia.
They have led the country in alternate terms since 1991 but have not spoken for years.
Parliament has been boycotted regularly by whichever party is in opposition, and a culture of street demonstrations has developed.
The BNP is calling for its supporters to be ready to take on opposition activists.
Our correspondent says that there will be little the 25,000 police and security personnel deployed in the capital, Dhaka, can do as the powerful political parties try to enforce their dominance of the streets.
With the elections not expected to take place until January, months of disruption lie ahead, our correspondent adds.
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