Young children across England are learning what school they will be attending this autumn, and nearly 90% will have been given their first choice.
However, early indications suggest wide differences around the country, with nearly one in six children missing out on their preference in some areas, compared with almost all getting their number one pick in others.
At least 24 local authorities came under the average of 88.9% for placing youngsters in their top choice of primary school.
A continuing squeeze on places - fuelled partly by a rising birth rate in recent years, combined with the effect of immigration in some areas - means that some parts of England are struggling to accommodate demand, it was claimed.
Initial results of more than 40 of the 152 local authorities from a Press Association survey shows that families in some areas are more likely to gain a place at their top choice than in others.
In Redcar & Cleveland, 97.8% of new starters have got their preferred place, along with 96% in Hartlepool.
Results from Liverpool show that 88% of children starting reception will go to their first choice, while Birmingham and Newcastle only placed 85.6% in their top slots.
Around 3.9% of children - 3,761 - were given a school that was not one of their preferred options.
Figures from Southampton show just 84.8% got their first choice - a drop of 4.1% - while in Trafford the percentage was 85%.
There has been, on average, a 12% rise in applicants - as official data shows that 3,807 primary schools are now "full or over capacity".
Staffordshire Council were pleased to announce that they had coped with the 11% rise in birth rate, managing to place 92.7% of Staffordshire children in their first choice.
County Councillor Ben Adams, Cabinet Member for Learning and Skills, added: "With an increasing population and new housing developments being built across the county, demand for primary and secondary school places means we have to work hard to ensure every child in Staffordshire has the choice to attend a good, local school.
"We estimate that over the next five years, we will need around 7,000 primary and 3,000 secondary school places to meet expected demand. By building new schools and making better use of existing accommodation, we can ensure the majority of parents are offered one of the top three preferred places."
Priti Patel, Conservative Eurosceptic employment minister, said it was "deeply regrettable" that families in England would be hit by a shortage in primary school places and claimed that "uncontrolled migration" puts pressure on public services.
She said: "The shortage of primary school places is yet another example of how uncontrolled migration is putting unsustainable pressures on our public services.
"Education is one of the most important things that Government delivers, and it's deeply regrettable that so many families with young children are set to be disappointed today.
"The truth is that for as long as we remain a member of the EU we are completely unable to control the numbers of people coming to this country - and with another five countries in the pipeline to join the EU the problem is set to get even worse."
However, a Whitehall source denied this and added: "There's no evidence that migration is the key driver of demand for primary school places."
Unions and town hall leaders warned Government reforms that mean all schools will convert to academies are set to fuel the shortage in school places.
Councils will not have the power to force schools to expand in the future, even where there is demand and capacity, the Local Government Association (LGA) said.
It warned that an additional 336,000 primary school places would be needed by 2024.
Roy Perry, chairman of the LGA's children and young people board, said: "Most academies will be keen to work with their local authorities, but in the minority of situations where this isn't the case, appropriate powers are vital to ensure all children get a suitable place."
Lucy Powell, shadow education secretary, said oversubscribed schools mean children are crammed into supersized classes.
She said: "This Government's broken school places system means that children are being crammed into ever larger class sizes and many schools are already at or over capacity.
"Ministers have already tied the hands of local areas to adequately plan for school places. The Tories' new forced academisation policy will make the school places system implode, as councils lose completely the levers they have to ensure there are enough school places for children."
The Government spent £5 billion creating places between 2011 and 2015 and 95.9% of parents received an offer at one of their top three preferred primary schools last year, the Department for Education said.
"Despite rising pupil numbers, at primary, the number of pupils in excess of their school's capacity has fallen by a quarter since 2010, and average class sizes have seen little change," a spokesman said.
"Of course there is more to do - that's why this Government has already committed to invest a further £7 billion to support councils in delivering school places, which along with our investment in 500 new free schools we expect to deliver 600,000 new places by 2021.
"It is simply not true to suggest councils cannot commission new schools - where councils identify that a new school is needed in their area they are required to run a competition to identify strong providers for a new free school."