Strictly speaking, ground is a connection to the Earth. When we speak about a "safety ground", it is this kind truly earthed connection.
In circuits, the "common return" path to the power supply is informally called "ground", even though it is not actually earthed. Battery-powered devices, and the electronics inside airplanes, still have grounds.
All circuits have to have return paths to the power supply. Ground is only a convention. The designer decides that a particular network connected to the power supply is at a 0V potential. Other voltages are measured with respect to this potential. But, of course electrons do not read schematics and do not understand what ground is, and what point is 0V. Moreover, conventional current is opposite to the flow of electrons: Benny Franklin got it wrong. So when the negative terminal is grounded, electrons are not actually returning to ground, but emanating from there.
If the power supply is a battery, and the devices in the circuit are designed for a positive supply, then the negative terminal serves as the ground. The positive terminal could also be ground, if all the circuits are designed for a negative supply.
"Designed for a positive supply" means that the key voltages in the circuit are positive, measured from the negative return. For instance, many NPN transistor circuits, such as an emitter follower, have a positive sense. The collector is connected to a positive supply, and the emitter branch of the circuit is on a network which is goes to a negative ground. The input signal is understood to be between ground and the transistor base.
However, the same circuit could be built using a PNP transistor instead, using a negative power supply, whose positive terminal serves as ground.
There also exists, in wide use, circuits that run on dual voltage power supplies. The power supply provides a positive voltage, a negative voltage, and a ground. The two voltages are usually approximately equal (but opposite) so the ground is nearly exactly in between them.
you are right, you oscilloscope gnd is connected to earth.
You battery can modeled basically by a floating voltage source of 12V.
Because it is floating, you can connect its minus terminal to the gnd of the oscilloscope.
At the positive terminal of the battery you will get a voltage that is 12V greater than your oscilloscope gnd. If now you want to measure that 12V with your oscilloscope probe, make sure that 12V is not greater than the max input voltage of the probe. This is generally not an issues if you use standard 10x,1Mohm probes.
But pay attention if you plan to connect to the oscilloscope directly using a BNC connector for instance. The Inputs of my Lecroy accept max 5V and are 50 ohm...
Imagine 12V in 50 ohm :
$$ P = \frac{U^2}R $$
$$ P = \frac{12^2}{50} = 2.88W $$
2.88W the be dissipated by the input of the scope...
Best Answer
The negative of the battery is connected to the chassis so run your postive to the battery (make sure it is fused) and connect the negative of your siren to the chassis. Look for any bolt nearby, use a ring terminal and fix it down. However do make sure that the chassis surface is bare metal (sand it down), any significant amount of paint will result in a bad connection and cause problems.