He remained in custudy on Tuesday evening.
Mr Snelling is believed to have worked for a mental health charity in York and, according to local residents, is an avid follower of Newcastle United.
His home in Burnholme Grove, near Burnholme Community College, York, was being searched by officers in forensic suits.
Police activity centred on the back garden and a paved area behind the garage.
Just after 1.30pm, officers removed an evidence bag and suitcase from the house. Another officer has been taking video footage of the house.
Officers were searching the house and an area behind a garage. The downstairs curtains were closed. The house is around three quarters of a mile from Claudia's home.
Neighbours said a police van arrived at 8am and then a larger van at 11.30am. They said they believed the occupant lived on his own. Camera crews are now outside the house.
Forensic examinations and searches were expected to take several days.
A car has also been seized as part of the enquiries.
Detectives said they would like to stress that Ms Lawrence's body had not been found.
Police said they could not rule out the possibility of further arrests in the future.
The scene at Burnholme Grove, York, this morning
Her family has been notified and are being supported by trained officers, the force said.
Neighbours said Mr Snelling had worked at York University as a lab technician until recently and he had lived in the house for more than 20 years.
Terry Ellis, who has lived in Burnholme Grove since 1961, said: “The first thing I saw this morning was when the police investigation car came up. I thought that maybe someone had broken in, because he’s had someone break in before.”
Jen King, one of Ms Lawrence's closest friends, said the arrest showed progress was being made with the investigation.
She said: "It has been five years. It proves it's very much still alive in the minds of the police and it just goes to show despite the accusations from people that the police are not bothered they are very much bothered about it.
"If it goes somewhere that's great but if not it shows the investigation is ongoing.
"It's a step towards this one day being resolved."
Claudia's mother Joan, who has worked tirelessly to highlight the case
Ms Lawrence's mother, Joan, said she found out about the arrest when police officers visited her home at 8am this morning.
“To be honest it hasn’t quite sunk in yet,” she said. “After five years and this happening so suddenly, it’s just not sunk in.”
Thursday, April 17, 2014: New people come forward in hunt for missing Claudia Lawrence
Thursday, March 20, 2014: Missing Claudia - fresh information in wake of Crimewatch appeal
Thursday, March 20, 2014: Ten calls to police following Crimewatch appeal about missing York chef Claudia Lawrence five years after she disappeared
Wednesday, March 19, 2014: Police reveal more new leads in the hunt for Claudia Lawrence
Wednesday, March 19, 2014: Three new appeals issued for people police want to trace over disappearance of Claudia Lawrence
Wednesday, March 19, 2014: New forensic methods provide leads after fresh examination of Claudia Lawrence's home
Wednesday, March 19, 2014: Police reveal new leads in hunt for Claudia Lawrence
March 18, 2014: Five years on from Claudia Lawrence's disappearance, her father speaks of "torture" of not knowing her fate
Darlington-born Claudia, then 35, a chef at York University, disappeared in 2009.
She was last seen after leaving work at Goodricke College, part of York University, on the afternoon of March 18, 2009.
After speaking to both parents separately on the telephone that night, she has not been seen or heard from since.
Despite one of the biggest investigations ever mounted by North Yorkshire police there has been no trace of her since.
CCTV image of Claudia Lawrence leaving work hours before she disappeared, March 18, 2009
A total of 121 people came forward to help the murder investigation after a fresh appeal for help on was made by police in March on BBC's Crimewatch.
More than 28,500 pages were viewed by more than 12,700 visitors on the police's new Claudia Lawrence microsite - northyorkshire.police.uk/claudialawrence
The new information was described as "encouraging and impressive, despite the five-year gap" by the head of the force's major crime unit, Detective Superintendent Dai Malyn, pictured below.
The appeals included a direct request for help to taxi drivers who may have dropped Claudia off in the Acomb area of York sometime before she disappeared or picked her up early in the morning.
Marking the fifth anniversary of his daughter's disappearnce earlier this year, Mr Lawrence described the strain of living through "18,000 days and nights" of not knowing why she had disappeared.
In another heartfelt plea for information, he said: "No one knows unless they are in the same position as we are what it means every day and most nights wondering, worrying.
"Claudia are you safe, is someone holding you, have you been hurt and even, of course, are you alive?" he said.
"It's now five years on and it is more than 18,000 of those days and nights.
"It's an awful long time for anyone to be in this position. It eats into you. It's like a cancer. It just gets worse and worse."
The disappearance of Ms Lawrence, whose parents lived in Darlington before moving to Malton, North Yorkshire, where she was raised, has become one of Britain'smost high-profile unsolved cases.
Police believe she was murdered, and finding out what happened to her became one of the first tasks of North Yorkshire's major crime unit when it was relaunched towards the end of last year.
Police outside the home of Ms Lawrence in Heworth, York, during a fresh review of the case last year
Anyone with information that could assist the investigation should contact North Yorkshire Police on 101, select option 1, and pass details to the force control room.
They should quote “Claudia Lawrence” when passing on details.
Information can also be passed to the independent charity Crimestoppers anonymously on 0800-555-111, quoting “Claudia Lawrence, North Yorkshire Police”.
Alternatively, go to www.crimestoppers-uk.org and fill in an Anonymous Online Form.