Severe or continuous bleeding is always an emergency.
Substantial blood loss can lead to shock, collapse, and death. This is no laughing matter. The smaller the dog and the faster the blood loss, the more your dog’s life is at stake.
There are, of course, situations when your dog might get a little nick or cut, where bleeding is minimal and stops on its own. Whether or not you should see a veterinarian with minor wounds depends on the type and depth of the wound. With an active dog, you might have learned how to treat minor wounds at home.
However, even a tiny nick in the skin from an animal bite can prove much more serious than it looks and it can blow up with a huge infection.
I recommend treating all animal bites as an emergency.
Further reading: Are Animal Bites an Emergency?
Blood in stool
A tiny amount of fresh blood in feces might not be an emergency. But it is a major red flag as it does indicate damage to the gut lining whether from infection, foreign material or inflammatory process. You might not need to seek veterinary care immediately but do it as soon as you can. Remember it could turn bad in a hurry such as with a foreign body or hemorrhagic gastroenteritis (note this one now got renamed to Acute Hemorrhagic Diarrhea Syndrome).
Further reading: Blood in Dog Stool
Blood in urine
Blood in urine or urine that has any color other than shades of yellow is a reason to seek medical help as soon as possible.
Further reading: Blood in Urine
Nose bleeds may or may not be an emergency depending on duration and the amount of bleeding but always call for veterinary attention.
Further reading: Should I Worry About Dog Nosebleeds? Does a Dog Nose Bleed Easily?
The bottom line is that there should not be any blood coming out of your dog anywhere.
While not every bleeding constitutes an emergency, meaning you need to see a vet immediately, they all do require veterinary attention.
The more severe the bleeding, the faster you need to get help.
That is, again good sense. If your dog is also lethargic, has pale gums, or showing other serious signs, time is of the essence.
First Aid for Bleeding in Dogs