Functions and Counting Problems in Streaming Computation I have read a stream computation paper in STOC07(Paul Beame, T. S. Jayram, and Atri Rudra. Lower bounds for randomized read/write stream algorithms.) and FOCS08（Paul Beame and Trinh Huynh. On the value of multiple read/write streams for approximating frequency moments).
I am interested in this topic with design of algorithms and analysis of limitations.

I want a list of natural counting functions, which stream algorithm theorists often consider like NPcomplete problems list by Garey and Johnson .

Is there such a list ?
I don't know if there's a list of canonical hard problems, but there is a list of open problems in streaming as maintained by Andrew McGregor. This is not limited to the specific models you're referring to  though.