If a cell phone is not hooked to internet, can listening software work on it?

My cell phone does not have an internet plan, but if you push the button, it will come on and they charge me.

Can listening spyware such as spybubble or flexispy still work on my phone? I am afraid someone has put one of these on my phone. Is there a way to find out through phone records, or data records if it has been put on my phone?

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