How to Test Crystal Oscillator with an Oscilloscope

https://www.testandmeasurementtips.com/measuring-clock-oscillator-frequency/

An oscilloscope will show you the waveform. A frequency counter is a better way to test the frequency of a oscillator. Some 'scopes will have a counter built in. Good luck with your studies.

If you are looking to measure the frequency accurately, a frequency counter is best, but it may give errors if it loads the oscillator circuit down. Connecting the counter to the low-impedance output of the driver is better than connecting it to the high-impedance input of the oscillator.

If you are viewing the waveforms with an oscilloscope, you can do a simple experiment to see if your scope probe is affecting the displayed waveform by touching a second probe to the same test point. If the displayed waveform changes, then the original waveform is probably distorted by the first scope probe. Adding a 1k resistor in series with the scope probes and repeating the experiment may show a more accurate waveform, possibly showing that the second added probe (and resistor) does not change the displayed waveform.

https://electronics.stackexchange.com/questions/219034/does-it-ever-make-sense-to-use-a-50%CE%A9-input-with-10x-scope-probe

Usually, the scope has an input impedance of 1 megohm resistance and about 20 pf capacitance. In order to get an accurate representation of the waveform at high frequencies, an X10 probe is used.

The X10 probe consists of a 9 megohm resistor and a variable capacitor. The probe is connected to the square wave output of the oscilloscope. When the probe is adjusted properly, the impedance of the X10 probe and oscilloscope series combination is 10 times the impedance of the oscilloscope alone, so the shape of the waveform shown on the scope (1/10 amplitude) is an accurate representation of the voltage waveform at the test point.

Yes, this picture shows the problem. The 20pF of the scope probe is the same magnitude as the 10-20pF specified loading capacitors for the crystal. At higher frequencies, this is a significant loading effect on the circuit. By adding a 1k or 10k series resistor, the loading effect is significantly reduced. At higher frequencies, there will be a voltage divider between the series resistor and the 20pF of the scope, so you shouldn't trust the measured amplitude, but you can trust the waveform shape and the DC level, since 10k and 10Meg will drop the voltage by less than 1%. You can guess the amplitude reduction by adding a second similar resistor and probe to the same point to see if the observed signal on the first probe is reduced by the double loading of the added second probe.

If you are dealing with frequencies of ay 100 khz or above, touching the 'scope probe to the circuit which can effect it's working (e.g. an oscillator that doesn't start, may well start when touched by the probe).
To overcome this you can wind say 6 turns of thin insulated wire into a coil (say the diameter of your thumb) connect one end to the scope probe and the other end to the probe earth clip. If you hold this over the circuit it will pick up the oscillations.
Del

You have not given us much clue. Xtal oscillators in common use vary from ~32.768 Khz in a wristwatch or clock to ~200 MHz metal box computer clock with square wave output.

Do you know supply voltage for module or is it in a unit with its own power? Do you have a frequency written on crystal or unit?

If it is low frequency, you may be able to pick-up output on a radio receiver. Connect oscillator output to 50cm of hook-up wire. Hold LW/MW/SW/FM ferrite rod/whip aerial radio close to wire & tune around on each range at max volume. You may get a sudden increase or fall in "hiss"; or a broadcast station with a "whistle" due to a beat with Xtal - turn off Xtal to be sure noise is not due to something else. Example: On MW a 100 kHz square output would appear at 100 kHz intervals on dial with similar strength. If the Xtal unit has sine wave output, any harmonics will be weak (they may only be generated in the mixer of your radio).

Connecting a germanium or schottky detector diode [not a power rectifier type, a 1N4148 or equivalent signal diode will do, but less sensitive] to common of Xtal output with other end of diode connected to "hot" Xtal output via 100 pF ceramic or mica capacitor. Connect DVM direct to common of Osc. and to diode/100pF junction via 100k to 1 megohm wire-end carbon/ metal oxide/film resistor. This will read peak voltage at above a few hundred kHz but less owing to 100 pF reactance if frequency is lower.