The average asking price remained at a "virtual standstill" in July - up by just 0.1% to £317,421.
The number of agreed sales in June rose by 4.6% year-on-year, while the number of sellers was up by 7.6%.
Prospective buyers are "seeing a lot of sold boards on properties they would like to buy themselves" with over 45% of estate agents' property stock now being sold subject to contract.
Miles Shipside, Rightmove director and housing market analyst, said: "A year on from the shock referendum result and subsequent dent in activity levels, the fundamentals remain strong. Low unemployment, low interest rates, strong demand and historic undersupply of homes are mitigating any wobbles in confidence and, as a result, almost half the properties on the market - over 45% - have sold signs slapped across them".
Murphy explained the United Kingdom property market is seeing the highest levels of stock sell-through since Rightmove started to track the metric - which is means demand for property remains strong with consumer confidence largely undaunted, and underlying market fundamentals remaining robust.
Rightmove said asking prices in most of England fell, but the overall average inched up due to gains in London contrasting with a sharp fall in June in the capital.
Property experts believe that buyers have hit their price ceiling as double-digit house price growth has far outpaced wages over the past five years.
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Growth has returned to the United Kingdom housing market this month, but house prices have edged up just £300, new figures have shown.
"Buyers, many of whom are sellers too, will struggle to afford to pay much more", Shipside was quoted as saying by the Daily Telegraph.
The North East saw the biggest month-on-month fall in asking prices, with a 1.3 per cent decline, followed by Yorkshire and the Humber with a 1 per cent fall and the North West and East of England, where prices fell by 0.7 per cent.
In the East of England asking prices increased by just 0.6% month on month to £351,276 but are still 3.8% higher than a year ago and in Wales they increased by 0.8% to £184,308 but are 0.5% down year on year.
"Quite to the contrary, all of these indicators would point to a calm, steady and functioning United Kingdom market, with perhaps those consumers who did decide to "wait and see" in the lead up the Election now deciding to simply get on with their move, adding yet more motivated movers into the mix in most parts of the UK".
Brian Murphy, Head of Lending for Mortgage Advice Bureau, commented: "Seeing the slow down on new instructions reported in the data is no great surprise at all, given that we're now entering peak holiday season, and this is the normal and expected pattern every year, so is no cause for concern whatsoever".