Eugene Volokh notes that Elena Kagan does express one clear opinion about a Supreme Court decision in the 105 pages of her 1996 University of Chicago Law Review article (PDF) on First Amendment doctrine (which I discuss here and here). She says the Court "made the correct decision" in United States v. Eichman, the 1990 ruling that overturned a federal ban on flag desecration. "I believe this is the only occurrence of the phrase 'correct decision' in her articles," Volokh writes.
The law at issue in Eichman was the congressional response to Texas v. Johnson, the 1989 decision in which the Court overturned a state ban on flag burning. The Texas law made it a crime to "deface, damage, or otherwise physically mistreat" a flag "in a way that the actor knows will seriously offend one or more persons likely to observe or discover his action." In an attempt to write a law that would survive judicial review, Congress phrased the federal ban in broader terms, without referring to the message communicated by flag descration. The federal law applied to anyone who "knowingly mutilates, defaces, physically defiles, burns, maintains on the floor or ground, or tramples upon" a flag. Kagan uses the case to illustrate her point that a seemingly neutral statute that makes no reference to speech can still violate the First Amendment if a motive to suppress certain kinds of expression is clear. "All of the rational interests underlying the law relate to the restriction of a message," she writes. "In such a case, an indicator of illegitimate motive (content-based justification) undermines the indicator of legitimate motive (general application)."
Notably, Kagan's endorsement of Eichman, which implicitly applies to Johnson as well, puts her at odds with the justice she would replace, John Paul Stevens, who dissented from both decisions. So on this free speech issue, she looks better than Stevens. It seems likely, however, that she takes a similar view of limits on the political speech of corporations (President Obama certainly seems to think so). And her 1993 article (PDF) offering advice to feminist opponents of pornography suggests she would be more inclined to uphold restrictions on material with sexual themes than Stevens was toward the end of his career. The same article also suggests she might be willing to uphold "hate speech" restrictions framed as general bans on harassment or "fighting words."
Addendum: As Tulpa notes in the comments, the federal ban on flag desecration included an exception for disposal of a "worn or soiled" flag, which suggests a concern about the expressive function of the prohibited acts. For that matter, the very notion of "desecration" (the term by which the offense is described in the statute) indicates the law was aimed at squelching a particular kind of speech.