Rapid planetary rotation can cause the equilibrium shape of a planet to be oblate. While planetary oblateness has mostly been probed by examining the subtle ingress and egress features in photometric transit light curves, we investigate the effect of oblateness on the spectroscopic Rossiter-McLaughlin (RM) signals. We found that a giant planet, with planet-to-star radius ratio of 0.15 and Saturn-like oblateness of 0.098, can cause spectroscopic signatures with amplitudes up to 1.1 m s-1 which is detectable by high-precision spectrographs such as ESPRESSO. We also found that the spectroscopic oblateness signals are particularly amplified for transits across rapidly rotating stars and for planets with spin-orbit misalignment thereby making them more prominent than the photometric signals at some transit orientations. We compared the detectability of oblateness in photometry and spectroscopy and found that photometric light curves are more sensitive to detecting oblateness than the spectroscopic RM signals mostly because they can be sampled with higher cadence to better probe the oblateness ingress and egress anomaly. However, joint analyses of the light curve and RM signal of a transiting planet provides more accurate and precise estimate of the planet's oblateness. Therefore, ESPRESSO alongside ongoing and upcoming photometric instruments such as TESS, CHEOPS, PLATO, and JWST will be extremely useful in measuring planet oblateness.