Basenjis don't bark but yodel and make other unusual vocalizations. They are quieter than most breeds.
French Bulldogs rarely bark without reason. They prefer snoozing on the sofa over being vocal.
Greyhounds are quiet dogs that reserve their "voice" for expressing happiness. They sleep most of the day.
Whippets bark minimally and make quiet whining sounds when they want something. They are laidback.
The Bernese Mountain Dog barks occasionally but usually only when alerting their owner or greeting people.
The Cavalier King Charles Spaniel rarely barks excessively though they aren't mute.
Pharaoh Hounds "blush" by barking and making other strange noises rarely. They are quiet dogs.