I really can't think of any influences of Scots Law on U.S. law. In all of the areas where I know Scots law to be distinctive vis-a-vis English common law, the U.S. has not followed the Scots law model, and I have never seen a Scottish precedent cited in a court opinion in the U.S. while I have seen many English ones cited (with one narrow exception noted below).
For example, while the Scots Law concept of a criminal trial verdict of "not proven" rather than "not guilty" has been discussed by academics and policymakers, to the best of my knowledge, no U.S. jurisdiction has adopted a "not proven" verdict option.
Two U.S. states experimented with non-unanimous verdicts, something that U.S. law has since disavowed, but so far as I know, this decision was not borrowed from Scottish practice. In Louisiana, it was a calculated strategy to make it easier to convict black defendants after slavery was abolished.
No U.S. jurisdiction has ever had petite juries of 15 in any kind of case.
Scotland and the U.S. are now aligned in being jurisdictions that have jury trials of ordinary personal injury cases, something that very few countries do, but in the U.S. case, the right to a jury trial in personal injury cases flows from English law practice that England later modified while the U.S. did not, rather than from copying Scots law.
Scotland still has many common law crimes, which U.S. jurisdictions are in the final stages of completely abolishing (most have done so, but there are a few partial stragglers). Similarly, U.S. criminal law, unlike Scots Law, has never required corroboration for a criminal conviction, although, of course, it is desirable to have it if possible.
The Scottish Court system has much more specialization of court jurisdiction than almost all U.S. jurisdictions do, although U.S. jurisdictions vary in the extent to which they have specialized courts, and of course, no U.S. jurisdiction has had religious courts since the early 1800s before the church was disestablished in New England.
While the U.S. merged law and equity for the most part, as did most common law jurisdictions, the distinction still has a significant residual effect that differs from Scots law which never fully separated the two. Equity is still taught as a distinct elective class in many U.S. law schools to this day.
Perhaps the only area I can think of where there has arguably been some borrowing from Scots Law has been in the jurisdiction and choice of law analysis applied to marriage and divorce questions. It used to be easier to get married in Scotland than in England, and in cases of parental or religious opposition, couples sometimes went to Scotland to get married and left courts to sort out whether that was legitimate after the fact. This is an area where U.S. case law sometimes referred to how those issues were resolved in the cases of disputes over whether English or Scottish laws related to marriage formation and marriage dissolution should apply.
I don't know the law of mineral rights well enough to describe the provenance of some of its doctrines accurately, but there are some similarities between the Scottish concepts applicable to the structure of mineral rights, when they are reserved in a patent from the sovereign of the surface rights to property (as many key mineral rights in Scotland were starting in the early modern period), that may have some Scots Law influence.
The lack of Scots Law impact on U.S. law despite significant Scottish immigration to the U.S. is to some extent a function of U.S. immigration history.
Scottish immigration that was early enough to have an impact on the legal system's larger outlines was mostly "Scotch-Irish" (i.e. Protestants from the Scottish borderlands, often traditionally herders or subsistence farmers, who were ancestral to the current Irish Protestants of Northern Ireland), who were lower to middle class, who engaged mostly in rural subsistence farming in Appalachia and the rural American South, and were not prominent in commerce and were subordinate to an English elite (the Episcopal Church of English origins has always a higher socioeconomic status church in the U.S. than the Presbyterian Church which has Scottish roots).
By the time middle class Scottish people engaged in commerce and inclined to participate in formal law and politics arrived in the U.S., a wave of migration reflected in the American West mostly from the mid-1800s as part of the gold rush and homesteading movements, the die was largely already cast with respect to major framework level issues in the U.S. legal system. The map below partially obscures this history to some extent, however, because many people with Scotch-Irish ancestry now identify for census purposes as "American."
Many anthropologists attribute the "culture of honor" of the American South and Appalachia to a Scotch-Irish ancestral source, and this culture of honor surely did influence the character of various U.S. legal doctrines and legislative developments in the U.S., but it did not involve direct borrowings from Scots Law, and indeed, it can be seen as deriving from a subculture of people for whom formal legal institutions in Scotland were sufficiently weak in practical application that they had little impact on their lives.
Honestly, if I had to hazard a guess, I would say that there have been more borrowings from Australian legal innovations (especially in probate and real property), and from civil law systems (for example, the law of good faith and fair dealing, and the law of unjust enrichment), than there have been from Scots law.
I would be happy to be corrected if someone is aware of a counterexample, however.