There is a bit of information around, and like @BryanKrause said, half-lives are a more useful measure of the life-time of the system rather than complete clearance, which can't be measured easily for something of this nature, due to the technologies we use to measure these sorts of things.
As far as I can tell, there isn't any information that has been released specifically about the persistence of the spike mRNA from the vaccine, because this is hard to measure. Normally with these sorts of things we measure a proxy for the target. There have been several papers that looked at persistence of mRNA and protein production from the mRNA to see how long it lasted in vivo and in vitro.
The easiest to understand is probably this paper1 (see fig 2C linked below), where they looked at luciferase (a light emitting protein) expression over time in mice injected with a mRNA. They show some production of light and hence luciferase expression to over 12 days, though note that's a log-scale on the y-axis, so <50% of maximal light (max of ~107 relative luminescence units) is <3 days, probably closer to 2 days. Conversely, this paper2 (see Fig 4C linked ->), found little expression 24 hours post-injection. They also found no translocation in the body, which is similar to most, though I can also find information that there seems to be translocation of signal to the spleen and liver (as one might expect), as well as other sites, but I'll come to that.
The best information isn't actually in the papers; it's in the Use Authorities from the medical governing bodies for various jurisdictions. I found one for the Pfizer/BioNTech vaccine3 (BNT162b2 - beware that's ~140 pg PDF) from the European Union with some interesting information. They say that the excipients (lipid particles) have a half-life of 140 hours (p. 46) in the blood following intra-venous injection, and T0.25 in the liver of 2 weeks, but also state that IM injection is similar.
A proxy for the spike, luciferase again was also mentioned:
Luciferase signals at the injection sites, most likely reflecting
distribution to the lymph nodes draining the injection sites, peaked 6h post injection with signals of approximately 10 000 times of buffer control animals. The signal decreased slowly during the first 72
hours and after 6 and 9 days the signals were further weakened to approximately levels of 18 and 7
times the signals obtained from animals injected with buffer control.
They also looked at distribution by radioactive labeling of the mRNA and looking to see where it went. They used 50 micrograms of the RNA for this, which is more than a dose for a person, but put into a mouse! They probably used so much to get greater sensitivity from their measurements.
Over 48 hours, distribution from the injection site to most tissues occurred, with the
majority of tissues exhibiting low levels of radioactivity.
Radioactivity was detected in most tissues from the first time point (0.25 h) and results support that
injections site and the liver are the major sites of distribution. The greatest mean concentration was
found remaining in the injection site at each time point in both sexes. Low levels of radioactivity were
detected in most tissues, with the greatest levels in plasma observed 1-4 hours post-dose. Over 48
hours, distribution was mainly observed to liver, adrenal glands, spleen and ovaries, with maximum
concentrations observed at 8-48 hours post-dose. Total recovery (% of injected dose) of radiolabeled
LNP+modRNA outside the injection site was greatest in the liver (up to 21.5%) and was much less in
spleen (≤1.1%), adrenal glands (≤0.1%) and ovaries (≤0.1%).
So, it would seem that the mRNA itself is cleared fairly quickly; probably the majority is gone within 3-4 days. But some will persist for some time, perhaps as much as 6 weeks, and it will spread to many sites in the body, mainly the liver and spleen.
Oberli, M. A., Reichmuth, A. M., Dorkin, J. R., Mitchell, M. J., Fenton, O. S., Jaklenec, A., Anderson, D. G., Langer, R., & Blankschtein, D. (2017). Lipid Nanoparticle Assisted mRNA Delivery for Potent Cancer Immunotherapy. Nano letters, 17(3), 1326–1335. https://doi.org/10.1021/acs.nanolett.6b03329
Karikó, K., Muramatsu, H., Welsh, F. A., Ludwig, J., Kato, H., Akira, S., & Weissman, D. (2008). Incorporation of pseudouridine into mRNA yields superior nonimmunogenic vector with increased translational capacity and biological stability. Molecular therapy : the journal of the American Society of Gene Therapy, 16(11), 1833–1840. https://doi.org/10.1038/mt.2008.200
19 February 2021
Committee for Medicinal Products for Human Use (CHMP)
Common name: COVID-19 mRNA vaccine (nucleoside-modified)
Procedure No. EMEA/H/C/005735/0000