Or are Equip activated abilities only usable after the Equip spell stack resolves and the equipment is a permanent (not a spell)?
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Can Leonin Shikari activate the equip abilities from instantly casted Equipment (by Sigarda's Aid)?
No; in general, you can't activate any ability of a spell (such as an Equipment spell's equip ability [C.R. 702.6a]) unless the ability states otherwise (e.g., Lightning Storm's activated ability) (C.R. 113.6).
Or are Equip activated abilities only usable after the Equip spell stack resolves and the equipment is a permanent (not a spell)?
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-MadMage
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