Two men, aged 32 and 39, and one woman, aged 55, were arrested on suspicion of theft and taken into custody at Gablecross police station.
The trio are believed to be linked to a large number of offences across the county.
Inspector Carly Nesbitt said: “These are significant arrests and come after we received a large number of reports of purse thefts up and down Swindon and Wiltshire.
“It shows how vital information from the public is. Someone saw our social media appeal and realised they recognised the people pictured and could assist with our enquiries.
“These three people will now be interviewed in custody and our investigation will continue.”
Advice on keeping purses safe was issued, including keeping hold of it on public transport.