Should I be following up after submitting an application?

Sorry in advance if this is a naive question.

I have been applying for jobs for a while and I'm rarely getting any response. I'm applying for jobs for which I meet or exceed the education and experience requirements. I typically get a system generated email saying my application has been recieved, but that's about it. I often don't get rejection emails. I would greatly prefer being told I was rejected than being left in the dark.

Should I be following up with the company? Maybe ask if I can send them some scripts I've made that I think demonstrate my suitability for this position?

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