If a person has dual citizenship (Canada and US), what will the foreign national potential spouse (from Europe) have to go through in order to be legally residing in the US? Would they even be allowed to be a resident and immigrate to the US or would it be Canada?
If a person has dual citizenship (Canada and US), what will the foreign national potential spouse (from Europe) have to go through in order to be legally residing in the US?
The process is the same as for the spouse of any US citizen. The fact that the US citizen holds another nationality is not relevant.
There are three possibilities, depending on where the couple plan to marry. See https://travel.state.gov/content/visas/en/immigrate/family/fiance.html.
If the couple plan to marry in the US, the potential spouse should get a K-1 visa. After the foreign national enters the US in K-1 status, the couple must marry within 90 days, whereupon the foreign national applies for adjustment of status to that of a lawful permanent resident. See https://travel.state.gov/content/visas/en/immigrate/family/fiance-k-1.html. While the K-1 visa is classed as a "nonimmigrant visa," the end result of the process is a green card for the foreign national.
If the couple plan to marry outside the US, then there are two possibilities to choose from after they are married. See https://travel.state.gov/content/visas/en/immigrate/family/fiance.html
One possibility is for the US citizen to sponsor the spouse for an IR-1 or CR-1 immigrant visa as the spouse of a US citizen. The CR-1 visa applies if the marriage is less than two years old, in which case the permanent residence will be conditional on the continuation of the marriage. In this case, all the processing is done through the consulate ("consular processing"), and the foreign national becomes a lawful permanent resident immediately upon entering the US. https://travel.state.gov/content/visas/en/immigrate/family/immediate-relative.html.
The other possibility is for the US citizen to sponsor the spouse for a K-3 nonimmigrant visa. Similar to a K-1 fiancé(e) visa, this allows the foreign national to enter the US and subsequently apply for adjustment of status to that of a lawful permanent resident. See https://travel.state.gov/content/visas/en/immigrate/family/spouse-citizen.html.
Would they even be allowed to be a resident and immigrate to the US ...
There are circumstances that would cause the US to deny the petition and refuse the visa, but absent any such circumstances, the answer is yes, they would be allowed to be a resident and immigrate.
... or would it be Canada?
The US will certainly not require the spouse to immigrate to Canada, as it has no power to do so. The US also won't deny an immigrant petition simply because the sponsor holds another nationality. As mentioned above, though, they might deny immigration for some other reason, in which case the couple might try their luck in Canada.
To sponsor the spouse for immigration to Canada, however, the dual citizen would probably have to move to Canada. Furthermore, if there are grounds for the US to refuse immigration, Canada would likely refuse immigration too, on the same grounds.