Families in England learn on Tuesday if their children have come to their preferred elementary schools.
About one in ten families missed their first choice last year – but 98% got their first three places.
Primary schools have added 636,000 extra seats since 2010 to meet the rising numbers – but this demographic bulge is now moving to secondary.
Paul Whiteman, the teacher's head, said securing a place feels "like a fight for the parents".
Over 600,000 families find out where they were offered a school in the fall.
Great regional differences
The national image of applications will not be known until June, but the chances of a first-choice place have improved in recent years – from 88% in 2014 to 91% in 2018.
However, last year about 2% did not receive an offer for their three main preferences or the schools they mentioned.
There are big regional differences every year. Authorities such as East Riding of Yorkshire, Northumberland and Rutland have more than 97% of families who have their first preferences.
However, the lowest success rates are in London: only 68% of families in Kensington and Chelsea and 77% in Camden received their first choice last year.
A population boom has put places under pressure – but that has peaked and this year's claim numbers may be on a downward trend.
Primary schools have been building additional classrooms over the past decade, with the number of students increasing by around 15% between 2009 and 2018 to 4.7 million.
The average primary school grew by 42 additional places, but these were not evenly distributed, with some growing very fast and some still having to cope with demand.
Mr. Whiteman, Secretary General of the National Association of Head Teachers, called for a closer "national strategy" to secure enough seats.
Otherwise, he said, "the annual anxious waiting for families will continue".
Mr Whiteman warned against a "random" expansion approach, so that "new places in the areas where they are most needed are not always commissioned".
The Minister for School Standards, Nick Gibb, said the standards had gone up and the primary education sector was "unrecognizable a generation ago."
He said 87% of elementary schools were now considered good or excellent, and the use of Phono lessons had improved the reading of children.
"In practice this means that even in cases where parents are not getting the news they were hoping for today, there is a likelihood that their child will attend a school that provides first-rate education," said Gibb.
But the New Schools Network, which promotes free schools, said that too many children were still attending schools rated "good".
"Finding out which elementary school your child will visit should be a time of excitement, and today nearly 100,000 families will find that their child is sent to a school that is not good enough," said group leader Luke Tryl.