Which interrupts are enabled when programming an ATTiny85 with Arduino IDE? I have programmed a tiny85 using the Arduino IDE. I am using the pulseIn() function. Wrapping it in noInterrupts()/interrupts(); appears to make the function more consistent. This would imply that interrupts are running.

Which interrupts are running? Can I disable them?

I am using the analogWrite() function to pins that have PWM support. Disabling the interrupts for a "long" time does not seam to effect these, this implies the PWMs are producing the analogue signal in hardware and so interrupts are not required.

(I know I can improve the code by not using pulseIn and using interrupts, however the question I have at the moment is which interrupts are running when I am not explicitly enabling any.)
The only interrupt that is always running is the one which generates the count for millis() and micros(). Disabling interrupts disables your ability to keep track of time and use delay().

Timer interrupts are also used by the tone() function and for sending / receiving serial data if the chip has a UART.

Disabling interrupts would also affect the use of external interrupts and pin change interrupts should you be using them.

It sounds, from your brief description, that you would only be effecting the use of millis() and delay() since you aren't using the other systems that use interrupts at all.