I tried to connect a "SparkFun SD card shield" (for 3.3V) SparkFun microSD Transflash... kaufen bei BerryBase
to the Arduino Mega (for testing purposes) and later to the Arduino Pro Mini 3.3V. I used the pins for the Arduino Mega: 50: DO (MISO) , 51: DI (MOSI) , 53: CS (ChipSelect), 52: SCK. VCC is 3.3 V and GND. I opened the test file "CardInfo" in the Arduino IDE and set the CS pin to 53. But I get an error message "My SD card is not recognized". I used an "Intenso UHS-I 16GB" (SDHC).
In theory, connecting the card reader direct to a 5V Arduino could have damaged it.
Try a much smaller SD card like 1 or 2GB or even less.
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All these SD cards need DOS FAT 8.3 file structure. Extended FAT and other versions are not supported.
I use THESE 32Gb microSD's without issue but they're 5v0's