My recollection is a little rusty - but here's what I remember. Mifare Classic is a 14443-1 RFID standard device. It uses passive tags, meaning that the tag/card does not transmit anything. When it is interrogated by the reader, it responds by load modulating its loop antenna which changes the amount of RF energy absorbed by the tag/card, thus changing the amount of energy reflected back to the reader. This is called backscatter.
The reader starts up sending an unmodulated 13.56MHz carrier for a few hundred microseconds - this allows the tag/card to harvest enough energy to "turn on". Then, it used ASK (amplitude shift keying) to talk to (interrogate) the card - this interrogation lasts about 1.5ms if I recall correctly. Then the reader returns to unmodulated 13.56MHz carrier. The tag/card responds a few hundred microseconds later with its backskatter reponse. This takes the form of a 847kHz subcarrier that is ASK modulated. I think the message length can vary, but 1-2ms duration is typical. Once the tag finishes its response, the reader usually turns off a few hundred microseconds later.
Thus, the total transaction is on the order of about 4ms.
Note that is should be easy to see the ASK modulation of the interrogator/reader, it might be very difficult to see the tag response because the amplitude change observed in the 13.56MHz envelope can be quite small. And, since the response is an amplitude modulated 847kHz subcarrier, the response might just look like noise, depending on how well you're coupled.