通販などで売られている健康器具で、腹、脚、腕などを振動させると、脂肪が落ちたり、筋肉がついたりするという説明がされていますが、エビデンスはあるのでしょうか？

どのような装置についてのご質問なのかは不明ですが、単に組織に振動を与えるだけで、脂肪が落ちたり筋が増えたりするという研究成果は見当たりません。
一方で、Whole Body Vibration (WBV)＝全身振動と言われる、30Hz程度の周波数で足下が1-5mm程度振動するような装置に立位で乗ったり、その上でスクワットなどの筋トレをすると、筋力、筋量、骨密度、バランス能力などが増える、あるいは効果がないという、無作為割付介入研究の成果が15程度あり（文献参照）ます。効果があるという研究の方が多いです。機械の上にただ立っているより、何か筋を刺激するような動作を加える方が効果が明確なようです。いずれの場合も高齢者のような筋力や低下した者での研究成果が多く、全ての人に効果があると言えるかは今後の研究が必要です。
必要であれば以下の文献をご一読ください。
運動ガイドラインプロジェクトリーダー　宮地元彦
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