Pottermore tells us that Azkaban was set up as a replacement for smaller prisons, suggesting that these were shut down when Azkaban became the main wizarding prison:
Once the International Statute of Secrecy had been imposed, the Ministry of Magic felt that the small wizarding prisons that existed up and down the country in various towns and villages posed a security risk, because attempts by incarcerated witches and wizards to break out often led to undesirable bangs, smells and light shows. A purpose-built prison, located on some remote Hebridean island, was preferred, and plans had been drawn up when Damocles Rowle became Minister for Magic.
Rowle was an authoritarian who had risen to power on an anti-Muggle agenda, capitalising on the anger felt by much of the wizarding community at being forced to go underground. Sadistic by nature, Rowle scrapped the plans for the new prison at once and insisted on using Azkaban. He claimed that the Dementors living there were an advantage: they could be harnessed as guards, saving the Ministry time, trouble and expense.
In spite of opposition from many wizards, among them experts on both Dementors and buildings with Azkaban’s kind of Dark history, Rowle carried out his plan and soon a steady trickle of prisoners had been placed there. None ever emerged. If they were not mad and dangerous before being placed in Azkaban, they swiftly became so.
And we know that minor offences such as Apparating without a licence are punishable by fines instead of incarceration, as seen in this answer, based on the following quote (emphasis mine):
"You have to pass a test to Apparate?" Harry asked.
"Oh yes," said Mr Weasley, tucking the tickets safely into the back pocket of his jeans. "The Department of Magical transportation had to fine a couple of people the other day for Apparating without a licence. It's not easy, Apparition, and when it's not done properly it can lead to nasty complications. This pair I'm talking about went and splinched themselves."
Harry had a sudden vision of a pair of legs and an eyeball lying abandoned on the pavement of Privet Drive.
"Were they OK?" he asked, startled.
"Oh yes," said Mr Weasley matter-of-factly. "But they got a heavy fine, and I don't think they'll be trying it again in a hurry. You don't mess around with Apparition."
-- HP and the Goblet of Fire