If you change the color of a spell which is to become a permanent, the permanent will retain the color change until the end of the turn.
But a mutate spell doesn't enter the battlefield.
If My opponent mutate Migratory Greathorn targeting his Grizzly Bears,
I response activating Blind Seer to make the Migratory Greathorn spell blue.
What color is the Grizzly Bears after the mutate spell resolves?
If Grizzly Bears is a legal target of Migratory Greathorn, Migratory Greathorn will leave the stack and merge with Grizzly Bears, but won't enter the battlefield as a separate permanent (C.R. 608.3b, 721.2b). If Migratory Greathorn is put on top of Grizzly Bears this way, Grizzly Bears will thus merge into a permanent with Migratory Greathorn's original characteristics (including, notably, the color green), not necessarily the characteristics Migratory Greathorn had when it was on the stack (under C.R. 721.2 and 702.139e, what is relevant are "the characteristics of its topmost component" of a merged permanent and, in the case of mutating, the abilities of that permanent's other components; C.R. 400.7; review C.R. 400.7a).Even so, the effect of Blind Seer's ability will "modif[y]" Grizzly Bears the same way as it "modifie[d]" Migratory Greathorn while it was on the stack (C.R. 702.139f). Thus, Grizzly Bears will be blue "until end of turn".
If Grizzly Bears is an illegal target of Migratory Greathorn, Migratory Greathorn will become a normal creature spell and enter the battlefield as a permanent, and Migratory Greathorn will remain blue "until end of turn" (C.R. 608.3b, 400.7a).
EDIT: Correct certain rule citations after comment 3 was posted; add another rule citation and a clarification after comment 3 was posted.
EDIT: Correction after comment 4 was posted.
400.7a Effects from spells, activated abilities, and triggered abilities that change the characteristics
or controller of a permanent spell on the stack continue to apply to the permanent that spell
becomes.
A strict reading of this rule would seem to support your position, peteroupc, as a mutating creature spell becomes no permanent. However, it is certainly counter-intuitive for a blue Migratory Greathorn on the stack, placed on top of a Grizzly Bears to be green. I wonder if this is in fact an oversight by the rules team rather than intentional design.
The merged permanent will still be affected by Blind Seer's effect, and therefore it will be blue no matter how you arrange its components. Effects that modify the mutating creature spell will keep modifying the resulting permanent.
Quote from CompRules »
702.139f Any effect that refers to or modifies the mutating creature spell refers to or modifies the mutated permanent it merges with as it resolves.
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But a mutate spell doesn't enter the battlefield.
If My opponent mutate Migratory Greathorn targeting his Grizzly Bears,
I response activating Blind Seer to make the Migratory Greathorn spell blue.
What color is the Grizzly Bears after the mutate spell resolves?
If Migratory Greathorn is put on top of Grizzly Bears this way, Grizzly Bears will thus merge into a permanent with Migratory Greathorn's original characteristics (including, notably, the color green), not necessarily the characteristics Migratory Greathorn had when it was on the stack (under C.R. 721.2 and 702.139e, what is relevant are "the characteristics of its topmost component" of a merged permanent and, in the case of mutating, the abilities of that permanent's other components; C.R. 400.7; review C.R. 400.7a).Even so, the effect of Blind Seer's ability will "modif[y]" Grizzly Bears the same way as it "modifie[d]" Migratory Greathorn while it was on the stack (C.R. 702.139f). Thus, Grizzly Bears will be blue "until end of turn".EDIT: Correct certain rule citations after comment 3 was posted; add another rule citation and a clarification after comment 3 was posted.
EDIT: Correction after comment 4 was posted.
A strict reading of this rule would seem to support your position, peteroupc, as a mutating creature spell becomes no permanent. However, it is certainly counter-intuitive for a blue Migratory Greathorn on the stack, placed on top of a Grizzly Bears to be green. I wonder if this is in fact an oversight by the rules team rather than intentional design.