To be able to program it with a USBASP you need to update it to the latest firmware:
PB0 needs a pullup resistor
USBASP | ATtiny10 |
VCC | VCC |
GND | GND |
MOSI | PB0 |
SCK | PB1 |
RESET | PB3 |
ATtiny10 Breakout Board with integrated status LED and a decoupling capacitor
If you enjoy programming the ATTiny10, check out my ATTiny10IDE project on GitHub at https://github.com/wholder/ATTiny10IDE/ In addition to many other features, my IDE can generate an Arduino Sketch that can program an ATTiny10. This lets you easily share your ATTiiny10 code with others that might not have USBASP programmer. I've also published plans for a simple High Voltage programmer you can use to recover an ATTiny10 that's had the RSTDISBL programmed.