Question 1 : When I'm taking a measurement with my scope, do I connect my scope's probes to the circuit before powering up the scope ?

Question 2 : If the scope and circuit are both powered up can I test anywhere in the circuit with the scope without causing problems ?

 

Given your questions, please watch this:

 

Question 1 : When I'm taking a measurement with my scope, do I connect my scope's probes to the circuit before powering up the scope ?

No.
The scope is normally powered first, and stays on for all measurements.

Question 2 : If the scope and circuit are both powered up can I test anywhere in the circuit with the scope without causing problems ?

There would be no problems, other than the oscilloscope probe resistive/capacitive load possibly affecting the measurement.
That's why a 10:1 probe is used to minimize such an affect.

 

Question 1 : When I'm taking a measurement with my scope, do I connect my scope's probes to the circuit before powering up the scope ?

The oscilloscope is turned on first and stays on all the time while making measurements. It would be a nuisance to have to turn the oscilloscope off and on again every time you need to take a measurement.

Question 2 : If the scope and circuit are both powered up can I test anywhere in the circuit with the scope without causing problems ?

Yes and No. It is safe to make a measurement as long as connections are made at the proper locations and you do not exceed the maximum voltage measurement rating of the oscilloscope.

More importantly, all voltage measurements are made between two points, the tip of the probe and the ground clip of the probe.
If you were to connect the ground clip to a node that is not ground something is going to blow!

 

The oscilloscope is turned on first and stays on all the time while making measurements. It would be a nuisance to have to turn the oscilloscope off and on again every time you need to take a measurement.


Yes and No. It is safe to make a measurement as long as connections are made at the proper locations and you do not exceed the maximum voltage measurement rating of the oscilloscope.

More importantly, all voltage measurements are made between two points, the tip of the probe and the ground clip of the probe.
If you were to connect the ground clip to a node that is not ground something is going to blow!

When I want to measure the p.d. across a resistor say, using a multimeter, with the red probe at the +ve end and the black probe
at the -ve end, I am effectively doing the same thing with a scope's +ve probes if I connect the scope's probe's +ve end to the +ve end of the resistor and the ground clip of the probe to the -ve end ?

 

When I want to measure the p.d. across a resistor say, using a multimeter, with the red probe at the +ve end and the black probe
at the -ve end, I am effectively doing the same thing with a scope's +ve probes if I connect the scope's probe's +ve end to the +ve end of the resistor and the ground clip of the probe to the -ve end ?

When you use a handheld multimeter, the meter is "floating", in other words, there is no reference voltage assigned to the red and black leads of the meter.

An oscilloscope is different (unless the oscilloscope is battery powered). The oscilloscope is already referenced to GROUND via the EARTH pin in the power plug. You can use the tip of the probe to take a measurement without connecting the ground clip. However, you will also be introducing extraneous noise into the measurement. In order to reduce noise pickup on the probe the ground clip should be connected to a ground node on your circuit under test.

If you attempt to measure the voltage across a resistor using the ground probe (as you did with the multimeter) one of these days something is going to go up in smoke (and that day could be today).

 

If you want to make oscilloscope voltage measurements between two points in a circuit where neither one is at ground potential, you must make a differential measurement. There are a couple of ways that you can do that. It is a simple measurement if you have a differential probe. You connect the two inputs to the points in the circuit, and the ground clip to circuit ground.
If you don't have a differential probe, but have a two channel scope, you can make the same differential measurement using two identical probes. The probe tips are connected to the points to be measured. The ground clips are connected to circuit ground. With both channels set to the same gain, set the vertical amplifier mode to A-B. If the scope only has A+B mode, simply reverse the polarity switch on one of the channels.