However, when programming, avrdude says:
avrdude: Device signature = 0x000000 avrdude: Yikes! Invalid device signature.
The command I use to program is:
avrdude -p attiny13 -c usbasp -U flash:w:main.hex:i -P usb
After googling for some time, I've realised I may have "locked" the AVR. I didn't pay attention to the output after first programming though. Is that possible I have locked the AVR if I didn't do anything related to fuses? Also, I've used this setup to program ATmega328P some time ago and it worked just fine.
And yes, I've used slow mode for the USBasp.