Last year saw 945 offences of failing to keep the animals under proper control, or contravention of an order, as well as 67 of failure to control, supervise or destroy, and 13 of dogs bred for fighting.
Although the total figure for 1,012 has fallen from a ten-year high of 1,292 in 2010-11, there are concerns it is not falling fast enough.
Last month, Broagan McCuaig, eight, was mauled by two American bulldogs in Glasgow.
She received serious wounds, requiring surgery to her face and leg, and suffered flashbacks after the attack.
Nanette Milne MSP, Scottish Conservative public health spokeswoman, who obtained the figures through a parliamentary question, said: “There have been a number of very high-profile cases.
“These have demonstrated the need for action to be taken to address the control and breeding of such dogs in Scotland.
“We need to promote responsible dog ownership and address the illegal breeding of dogs, and many charities have also expressed concern about this.”
She wants to see a review of dangerous dog laws.
The Scottish Conservatives’ call has been backed by an animal welfare charity.
Ian Robb, chairman of Angus Dogs Rescue, said: “Over the past four years, I have been trying to press the SNP Scottish Government into acting to address this growing problem.
“The number of incidents of dangerous dogs being abandoned has increased significantly with charities like Angus Dogs Rescue having to pick up the pieces.
“I have written to the First Minister to seek a meeting. However, this has not been forthcoming and the Scottish Government does not seem focused on trying to bring forward real solutions to address the problem.
“It is only a matter of time before another serious incident, or a death, will happen if this is not acted upon.”
Christine Grahame MSP launched legislation in February 2011 making owners responsible for dangerous and out of control dogs.
She believes this needs to be given time and also favours laws south of the Border, where pitbull terriers and three other breeds have been banned.
“Control of dogs focuses on the owners of dangerous dogs; it doesn’t just ban four breeds. That’s not the way to go.
“The Control of Dogs (Scotland) Act applies in premises, which the English legislation does not. And a lot of attacks take place in the home.”
The Scottish Government said it would be consulting on microchipping, but insists its laws are robust.
A spokeswoman said: “Scotland has a tough approach to dealing with dangerous dogs which focuses on preventing dog attacks in the first place.
“In contrast to England and Wales, our system gives local authorities powers to impose dog control notices with added conditions like microchipping and muzzling. Breach of these notices is a criminal offence.”