Reading Mifare Classic chips with Samsung Galaxy S4?
No, the S4 can't read MIFARE Classic cards. The main problem is that NXP does not license the reader-side of its proprietary MIFARE Classic technology (specifically the use of the (broken) Crypto-1 algorithm) to other chip manufacturers. As a result, the Broadcom NFC controller inside the S4 does not support communication with such cards.

This alone would not be too difficult to circumvent. The Crypto-1 algorithm has been reverse-engineered and published, so it could easily be implemented in software within an Android app. However, the MIFARE Classic protocol does not fully comply to ISO/IEC 14443-3 (NFC-A) in that the authentication command uses a non-standard frame format. As a result, the communication facilities accessible on Android (the closest would be the NfcA tag technology, that requires adherence to ISO/IEC 14443-3 framing) can't be used to communicate with MIFARE Classic tags.