transient analysis with square wave
Since you are interested in studying a transient response, what you need to do is apply a stimulus and observe the system (your LCR circuit) as it evolves to its steady state. When doing this theoretically, the two most suitable stimuli are the Dirac's delta function/pulse and the step function, because both apply some forcing quantity in zero time and let you calculate how the system evolves immediately after such stimulus has reached its own steady state (back to zero for the pulse, some positive value for the step). Those are called pulse and step responses, respectively.
Instead, when doing this experimentally, the easiest thing to do is apply a step (the steeper, the better), just because good pulses are somewhat harder to generate. With a common laboratory setup, the best thing you could do is to apply a square wave (which, after all, can be interpreted as a sequence of steps of opposite polarities) and observe the periodic result on an oscilloscope, so that you can see your LCR circuit response at every edge of your square wave. Of course, you need to take care that the period of the applied square wave is long enough to let the system reach its steady state before receiving an opposite stimulus.
Now, back at the core of your question, from the above you can see why a sinusoidal wave is not suitable for such a study: Since a sinusoidal wave is constantly changing, it does not allow the system to reach its steady state; it continuously forces the system to change as it tries to keep up with the sinusoid. Still what you would observe and how well your LCR circuit follows the stimulus depends on the frequency of your input sinusoid.
EDIT: As pointed out in the comments regarding the above strickenout sentence, a couple of remarks are required.

The term &quot;steady-state&quot; that I used is not correct because it indicates an equilibrium condition when the system continuously evolves forced by the input but transients have already vanished. This can be done with sinusoids (even though there exists a steady state to a square-wave input too); indeed, frequency-domain characterisation methods like Bode diagrams apply input sinusoids and measure the output amplitude and phase (relative to the input) at various frequencies.
However, the point of your question is to see and measure the transient behaviour of your system. This is usually done with a square wave input because it provides a periodic step input which allows you to observe a &quot;piece-wise transient&quot; (in the words of Vicente Cunha) output where, if the square wave frequency is low enough for your system, the responses to single edges have enough time to settle and transients have enough time to vanish. As mentioned in the comments, this kind of &quot;piece-wise analysis&quot; cannot be done with sinusoids.