A bit about the town. There are some waterfront communities in Becket that consists mostly of out of state owners using the homes as vacation properties. Many year round owners prefer to be out of the urban areas and owning lots of land. There is a big 'camp' community - several children's summer camps, a campground and a few stores. It's close proximity to the Mass Pike makes Becket a great location to get to.
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According the the official website of the MassachusettsDepartment of Fish and Game, www.mass.gov/dfwele, trout fishing is found:
Hoosic River in North Adams and Williamstown
Housatonic River in Great Barrington, Lenox, Pittsfield and Stockbridge. There has been extensive PCB remediation in Pittsfield and there are concerns about the Housatonic in Lenox. It is not recommended to consume fish from the Housatonic.
Laural Lake, Lee
Lake Buel, Monterey
York Lake, New Marlborough
Big Pond, Otis
Onota Lake and Pontoosuc Lake in Pittsfield
Windsor Pond, Windsor
Richmond Pond, Richmond
Stockbridge Bowl, Stockbridge
Goose Pond, Tyringham
There are numerous streams in Adams, Cheshire, and Savoy that support trout, but you'll need the goodwill of a local fisherman to find them!
None of these place are in Sheffield, but Stockbridge, Great Barrington, Monterey, New Marlborough, Otis and Tyringham are all relatively close by.
I hope this helps.
You may find additonal info on www.mass.gov/dcr/watersupply/lakepond/lakepond.htm.