Bloodhound by Tamora Pierce

Beka Cooper is finally a Dog, a full-fledged member of the Provost's Guard, dedicated to keeping peace in Corus' streets. But there's unrest in Tortall's capital. Counterfeit coins are turning up in shops all over the city, and merchants are raising prices to cover their losses. The Dogs discover that gamblers are bringing the counterfeit money from Port Caynn. In Port Caynn, Beka delves deep into the gambling world, where she meets a charming banking clerk named Dale Rowan. Beka thinks she may be falling for Rowan, but she won't let anything— or anyone—jeopardize her mission. As she heads north to an abandoned silver mine, it won't be enough for Beka to be her usual "terrier" self. She'll have to learn from Achoo to sniff out the criminals, to be a Bloodhound....