There are several commonly-used strategies for confirming specificity of a Morpholino oligo. These are the five-mispair experiment, the second non-overlapping 5'-UTR oligo experiment (and the similar two oligos against one mRNA experiments), the mRNA rescue experiment, and my current favorite, the Morpholino-in-a-null-mutant experiment.
I do not recommend the five-mispair control though its use has been commonly reported. Using a five-mispair oligo allows you to do an experiment to determine the effective-and-specific concentration range for your targeted oligo. The targeted oligo and the five-mispair oligo are used in side-by-side experiments. Each oligo is injected over a concentration range, with a group of embryos injected at each concentration. Usually there is a concentration at which the embryos injected with the targeted oligo display a morphant phenotype and the embryos injected with the five-mispair oligo appear wild-type; at a higher concentration, the embryos injected with the five-mispair oligo also display the morphant phenotype. We define the range of concentration between the onset of morphant phenotype with the targeted oligo and the onset of morphant phenotype with the five-mispair oligo as the effective and specific concentration (dose) range. In later experiments, targeted oligo injected within that range should produce specific effects, at least specific enough that if the targeted oligo shares some complementarity with a targetable region of an off-target mRNA but less than the targeted oligo's similarity with the five-mispair oligo then the targeted oligo should not knock down that off-target mRNA. However, it is not always the case that there will be an effective-and-specific concentration window for a given targeted-and-mispair set of oligos. Because of this, many investigators have shifted to using two non-overlapping oligos to demonstrate specificity.
The two non-overlapping 5'-UTR oligos experiment involves comparing the phenotype induced by injection of two different oligos targeted to block translation of the same mRNA. If both sequences induce the same phenotype, that supports the hypothesis that the observed phenotype is due to knockdown of the targeted gene. This has become a very commonly used test of specificity in the zebrafish community.
A variant on this experiment is to use a splice blocking Morpholino to produce the same phenotype as the translation blocking oligo; while this is a nice experiment when it works, it is not always easy to determine which exon to target in order to knock down the activity of the protein and phenocopy the translation blocker's effect. Yet another variation is the two-splice-blocker experiment, in which the same exon is targeted in separate experiments by a splice donor blocker and a splice acceptor blocker. If both of the oligos when used individually cause a clean exon excision, the phenotypes induced by the two oligos should be identical and, again, the hypothesis that the phenotype was triggered by specific excision of the exon is supported. However, activation of a cryptic splice site can confound the experiments using splice-blockers so while identical results indicate specific knockdown, dissimilar results do not preclude that interaction with the pre-mRNA was specific.
Another useful approach using two non-overlapping oligos is to coinject the oligos at reduced doses, so that each oligo is injected at well less than half of the dose which is just sufficient to elicit an altered phenotype when the oligo is used alone. If the coinjection at reduced doses elicits the same phenotype as the higher-dose single-oligo injections, this shows dose synergy and is support for the hypothesis that the phenotype is caused by knockdown of the targeted mRNA and not an unexpected RNA. This is proposed in: Bill BR, Petzold AM, Clark KJ, Schimmenti LA, Ekker SC. A primer for morpholino use in zebrafish. Zebrafish. 2009 Mar;6(1):69-77.
Note that coinjection of oligos targeting the star and guide strands of an miRNA may result in Morpholino dimer formation, as these oligos often have considerable complementarity with each other.
The mRNA rescue is an excellent proof-of-specificity experiment but will not work for some genes. For this experiment, an mRNA is injected which codes for the same protein that the Morpholino oligo knocks down, but the rescue mRNA has a modified Morpholino binding site so that the Morpholino target is not present on the rescue mRNA. For many genes, coinjection of the proper concentration of Morpholino and rescue mRNA results in producing a wild-type embryo, while injecting the Morpholino alone results in a morphant phenotype. However, the timing of the onset of translation is critical for some developmental genes and the early onset of translation resulting from co-injection of Morpholino and rescue mRNA in the early zygote may mess up the developmental process so that the embryos never recapitulate the wild-type phenotype. The location of translation can also be critical, but mRNA injected into an oocyte ends up distributed throughout the embryo.
That said, it is a very satisfying control when it is successful. If you are planning on using mRNA rescues, I recommend that you have your Morpholino targeted in the 5'-UTR without extending into the coding sequence. Some folks have tried rescuing Morpholinos targeted to the coding sequence by taking advantage of the degenerate genetic code to design mismatches into their mRNAs which do not change the amino acids encoded by the rescue mRNA. While this strategy makes sense, we know of several labs that have struggled with the technique without success. We think the chance of success is better if you start with a 5'-UTR Morpholino so that you can retain the original sequence of the coding region and use an irrelevant 5'-UTR sequence for your rescue mRNA.
If you have or can easily make a null mutant, that is now my favorite specificity control. When you put a Morpholino into a genetic null for the oligo's target, you should not see any effect of the oligo (that is, if microinjected into an early zygote, the oligo solution should have the same effect on development as injecting the vehicle solution without the oligo). If you do see additional effects, there are two possibilities: either the oligo is interacting with off-target RNA and causing the effects, or the mutant is not a true null. The Morpholino should also be injected into wild-type zygotes. It is unsurprising if the phenotype of the mutant and the phenotype of the Morpholino-injected organism differ; genetic compensation (change in expression of functionally-related genes) can hide the effect of losing function of a gene, as shown by Didier Stainier's group; a Morpholino can show the effect of the loss-of-function without the confounding effect of genetic compensation (Rossi et al. 2015). By avoiding cmpensation, the test of the Morpholino in a wild-type background can reveal gene function that would be obscured if only studying the mutant.
Rossi A, Kontarakis Z, Gerri C, Nolte H, Hölper S, Krüger M, Stainier DYR. Genetic compensation induced by deleterious mutations but not gene knockdowns. Nature. 2015 Aug 13;524(7564):230-3. doi: 10.1038/nature14580. Epub 2015 Jul 13.
Which one to use? That depends partially on what you have at hand. I would choose the easiest path first based on your resources. For instance, if you have made a CRISPR mutant, start with injecting the oligo into the mutant. If you can easily make a rescue RNA, try that (but don't be surprised if ectopic expression makes the rescue fail -- try some RNA in an embryo not treated with Morpholino to test the response to ectopic overexpression). If the easiest and least expensive path is to order a non-overlapping Morpholino, that is a reasonable path too. Think about your specific biological system and target gene - can you see a clear advantage to one of these approaches for your system? Might a report of the outcome of a particular specificity control add more value to your publication?