Since you posted this with the tag RNA-seq I will assume that you have RNASeq data. Since you mention transcripts I will also assume you have transcript level quantifications (else you can read more about how to obtain such transcript level quantification here.). I will also assume that you have at least two conditions (since else there is rarely a reason for doing RNASeq).
Assuming I am right there are 3 types of splicing analysis you can do:
Type 1: Exon based analysis:
The idea is simply that you analyze one exon at the time an see if an exon is differentially used (compared to all other exons in that gene). According to amongst others this article the best tool for doing this is DEXSeq (Bioconductor page, article link) which also allows visualisation of the changes. This is a powerfull way of analysing the data but can be hard to interpret the results.
Type 2: Splicing based analysis:
The idea with this type of analysis is to look at each splice event (exon skipping, alternative donor/acceptor etc) one at the time and see if there are systematic changes between conditions. The better tools for this is rMATS (as also mentioned by @MatthewP) and SUPPA2. This type of analysis is easier to interpret from a splicing perspective but harder to draw biological conclusions from.
There are also an extension of this analysis type which looks at groups of splice-events and detect changes within that group. This can be done with tools such as LeafCutter (github, article as also mentioned by @Prakash) or MAJIQ (more info here) these tools typically give more power but are even harder to interpret (except for something with splicing changed).
Type 3: Transcript based analysis:
The idea with this type of analysis it to utilize the transcript level quantification of the RNASeq data to detect changes in which transcripts are used in the two conditions (isoform switches). Although this is from a computational point harder (less events are found) the biological interpretation is a lot easier because you know the entire transcript. According to this paper again DEXSeq (adobted to transcript expression) seems to be the best tool to find such changes.
For the biological interpertration of such isoform switches I have created an R package called IsoformSwitchAnalyzeR. IsoformSwitchAnalyzeR enables identification and analysis of alternative splicing as well as isoform switches with predicted functional consequences (such as gain/loss of protein domains etc) from quantification by Kallisto, Salmon, Cufflinks/CuffFiff, RSEM etc.
IsoformSwitchAnalyzeR allows for analysis and visualizations of both individual genes as well as genome wide analysis of changes in both splicing and isoform switch consequences. You can see examples of the analysis types available here. For more info you can see how the analysis of switch consequences can be used in this article and for more info on the genome wide analysis take a look at this paper.
Please dont hessitate to ask if anything was unclear.
I would like to add one more tool recently out is Annotation-free quantification of RNA splicing using LeafCutter
You have a choice whether to make new transcript models or not, this depends on whether you want to try and find new transcripts. Otherwise you can just use the latest annotations in the ensembl gtf file. When you have worked out which way to go, then there are a few different pipelines one can use. Some tools do transcript level DE, others do exon level DE, both come at the splicing question from a slightly different direction. Here are some workflows:
HISAT2, stringtie, and ballgown # generates new transcript models and does transcript level DE
HISAT2, stringtie, and DEXSEQ # generates new transcript models and does exon level DE
HISAT2 and DEXSEQ # original transcript models and exon level DE
Salmon/ Kallisto and sleuth # original transcript models and transcript level DE (could generate a new transcriptome here as well)
I think there are a lot of good suggestions, but here is one more option that I don't believe has been mentioned yet:
While I think this is arguably a less popular option, it is actually my preferred choices for initial analysis. You need replicates, but I think that is even more important for junction/exon analysis (than gene analysis)