Most of the discussions about access to Scottish citizenship after independence have been about expat Scots and their descendants (as well as the related discussion about rUK citizenship for Scots).
However, I believe there is a bigger problem closer to home, concerning those who can take part in Scottish Parliament elections (and because of that also in the independence referendum), but who will lose that vote after a Yes victory, namely EU (and possibly Commonwealth) citizens resident in Scotland.
The SNP's 2002 proposal for a Scottish constitution (PDF) suggested granting Scottish citizenship to everybody living in Scotland on independence day ("Every person whose principal place of residence is in Scotland at the date at which this Constitution comes into force shall be a citizen of Scotland"), but the white paper states that only UK citizens will be Scottish citizens from day one, and that migrants are restricted to applying for naturalisation after independence, and only if they’ve lived here for at least ten years and are of good character (see the table at the end of this chapter).
This means that there’ll be a significant group of independence referendum voters who will effectively disenfranchise themselves by voting Yes in September. Some of them will just apply for Scottish citizenship afterwards, but a large group won’t qualify or will have their own reasons not to do so.
For instance, Denmark and several other EU countries don't allow dual citizenship to be acquired. It's allowed if you're born with two nationalities, or if you can get another one without applying for it (this typically happened in the past in some countries where wives automatically got their husband's nationality on marriage), but if you apply to become a citizen of another country, you lose your Danish citizenship. In other words, it wouldn't cause any problems if Scotland granted Scottish citizenship to all Danish citizens living in Scotland on 24 March 2016, but if they have to apply for naturalisation, they will lose their Danish nationality in the process.
At the moment, EU citizens living in Scotland can vote in all elections, apart from the Westminster ones. However, after independence Scottish Parliament elections will not be local elections any more, and EU citizens cannot vote in general elections in any other country, so I'd be surprised if Scotland was an exception. (The UK currently lets Irish and Commonwealth citizens take part in Westminster elections, but most other countries restrict voting in general elections to their own citizens.) If there are plans to let all EU citizens resident in Scotland vote in general elections after independence, please do let me know!
The result of this is that EU citizens living in Scotland are likely to lose their right to vote in Scottish Parliament elections when Scotland becomes independent. This is hardly a great incentive to vote Yes.
I strongly believe we can maximise the foreign-born Yes vote by granting Scottish citizenship to everybody who can take part in the independence referendum, not just to British citizens living in Scotland.