I've always thought it very useful with dogs that are light enough to be
picked up to have a safe method of doing it. It's important that the dog
is used to it and knows what to expect, so it needs to be practiced.
My method:
Squat down but keep your back straight.
Assuming you are facing the left side of the dog, so its head is to your
left. Put your left arm under the dog's chest with your hand facing
upwards. This means turning your arm anticlockwise. With your left hand
hold the dog's front right leg quite firmly just below the shoulder. At
this stage it's awkward for you because your arm is turned as far
anticlockwise as far as it will go. Simultaneously sweep your right arm
leftwards towards the dog's bum, holding down the tail. Your arm will
then be supporting the dog at the top of the back legs/bum area.
Stand up keeping your back straight, and lifting the dog. Lift the front
of the dog higher than the back. That allows your arm to rotate
clockwise to a more comfortable position. The dog's weight should be
roughly the same on both of your arms. The dog should be comfortable
because its weight is on naturally load-bearing parts: the bum/back legs
and the front shoulders, rather than on the chest.
I don't know anyone else who lifts dogs this way but it works for me.
When I took Bella to the vets he was surprised at how easily I lifted
her onto the table and how calm she was about it. It does depend very
much on the dog being used to it though. My old labs used to come to me
when we reached an obstruction on a walk and ask to be lifted over. I
couldn't lift a lab now though! Bella is just about capable of climbing
in and out of the van but will scrounge a lift if she can get away with
it. (She tried to be a stowaway the other day; attempted to jump into
the back of the van when I wasn't looking. It ended ignominiously for her.)

Bill