Does the mRNA of the covid19 spike protein contain any nuclear localization signals

Does the mRNA of the covid19 spike protein contain any nuclear localization signals?

I take it you are talking about the coronavirus SARS-CoV-2, and asking whether its RNA genome contains any nuclear localization signals that act post-translationally as signals to specifically import the spike proteins into the nucleus. I cannot find any reports where any of the spike proteins are nucleus-imported, i.e. the spike proteins do not have any (working) nuclear localization signal as far as we can tell.

Some of the proteins coded within its RNA genome are however imported into the nucleus. These are not the spike proteins though.

If you are interested in where the proteins localize within the infected cell, I suggest the following article as a very excellent starting place: A systemic and molecular study of subcellular localization of SARS-CoV-2 proteins

As for the mRNA, I cannot find any evidence that it localizes to the nucleus. However, there have been reports that other coronavirus' mRNA can localize to the nucleus, that CoV-1 contains a nuclear export motif, and that other coronavirus RNA can localize at the nuclear pore complex to interfere with trafficking between cytoplasm and nucleus space (i.e. at the gateway into the nucleus, but not inside!). This has been shown at least once more. However, these are not the spike proteins, but rather proteins like Nsp1 (non-structural protein 1) that are not categorized as spike proteins.

There has also been an interesting pre-print floating around since 2020 about the possibility of SARS-CoV-2 integrating into the human genome if we were to ectopically add a reverse transcriptase (RNA -> DNA writing enzyme) to the equation. Please read it with a grain of salt, I don't know that the paper has been peer-reviewed and unfortunately I cannot comment right now on the veracity of the research. It is also an in vitro experiment performed under circumstances that do not exist in humans, so the conclusions do not apply to discussions on the COVID-19 pandemic; it is only a proof of concept of integration-ability of the virus, rather that a demonstration that that occurs in vivo.

Here comes the cognate question:

Will we ever find the spike protein-coding sequences integrated into our genome?

A protein being inside the nucleus cannot integrate into the DNA genome though. With RNA, you would also require a reverse transcriptase to integrate it into DNA, which humans do not have. Therefore the odds of finding a spike protein-coding sequence in the genome, especially in the germline, if you are concerned with cross-generational effects, is virtually and practically zero.

EDIT:

You additionally ask in your edit whether DNA in the AZ and J&J vaccines could get incorporated into the genome (integration events). This is a known concern as it leads to carcinogenesis. You specifically ask about adeno-associated virus vectors. Below I quote an excerpt from Ura et al. (2014) Developments in Viral Vector-Based Vaccines from part 3.3. Adeno-Associated Virus Vectors:

Generally, recombinant AAV vectors are generated by deletion of the Rep and Cap coding regions between the ITRs. These regions are used for endogenous transgene expression. Owing to the deletion of these regions, AAV vectors cannot integrate into the host genome, and their DNA also persists in an episomal form. This preferable feature in the AAV vectors boosts their safety profile, by preventing the onset of tumorigenesis.

In other words, the virus is recombined and its ability to perform genome integration is taken away prior to any considerations of use as vector for the deliver of genes or immunogenic particles, such as the ones you would see in CoV-2 vaccines.

The first clinical trial of a therapeutic retroviral vector took place in 1990. Subsequent clinical studies have raised serious concerns regarding genotoxicity, mainly due to possible viral genome integration. The AAV vector has the ability to express episomal genes without integrating itself into the host genome, and has hence been approved by the EMA for clinical use.

As you can see, this has been in top consideration for a couple decades now, it is of course not an issue likely to be overlooked by regulatory bodies nor researchers or vaccine-producing companies.