Distinguishing Splicings that Produce Functional Transcripts Amongst Alternates Shown This is a link to the UCSC Genome Browser showing a portion of Ch 17 containing TP53 here.

There are several representations of the gene of varying length and number of exons. I presume these are equivalent to alternate splicings.

Assuming this presumption is correct(?), how can I determine which result in RNAs that are experimentally verified?
There are two resources worth checking: GTEx portal and Ensembl.
Uniprot and NCBI will call "canonical" the longest sequence comprising valid exons. These are not necessarily the biologically valid ones. 

According to Ensembl (TP53, there are a bunch that are experimentally validated and are all class 1. Note the Uniprot column being nearly homologeous: to match the transcripts one has to look at the length, in this case TP53-201 is 393aa long, which is class 1. As mentioned this is not always the case.

So which are more expressed in tissue X?
In GTEx, under the collapsed div labelled "Exon Expression" press the "isoform Expression" tab. The most expressed is ENST00000620739.4 (354aa long) join with ENST00000269305.8 (393aa).

In the Exon Expression you will find out what exons are the issue.

Now. to get the sequence of a transcript or covert a variant Ensembl drops the ball. There is no clear web tab that I know of. Nor an API route.
ftp://ftp.ensembl.org/pub/release-99/fasta/homo_sapiens/pep/Homo_sapiens.GRCh38.pep.all.fa.gz contains all the sequences however. For a project of mine (ENSTMapper in this link) to map an amino acid mutation on one transcript to another I ended up write my own short converter, although there are a few similar tools out there.