It depends on the circumstances and the landlords. This is a real grey are under tenancy laws. Generally speaking however, the landlord in under no real obligation to provide pest control, particularly if you have pets and/or you have a pest clause in your tenancy agreement which stipulates you will fumigate when you leave. You can always ask them, and they may oblidge, or not. If you are getting it done yourself make sure you do your homework and go through a place that has a 6mth-1yr garauntee (they will redo for free if the problem reoccurrs within that period).

For our Investment Property here in Canberra, I would consider pestcontrol IF there was a serious ****roach problem: but only if there was genuine evidence of a serious problem. I would not consider doing it for spiders, as it would be a pointless expense (canberra has a serious spider problem everywhere and it is impossible to completely erraticate them).