Firstly, here are some good reads on Microsoft Docs to help comparing the options available and make decisions based on your requirements:
(I mean requirements like Hosting features, Service Limits, 3rd party software installation and RDP access is required or not, Network isolation to a separate VNET is required or not, Cost considerations, minimum SLA, Regions available, instant deployment and auto-scaling, state management etc.):
- Azure App Service, Virtual Machines, Service Fabric, and Cloud Services comparison
- Decision tree for Azure compute services (This one covers a big spectrum.. simple virtual machines, Batch, Functions, Containers, AKS, ServiceFabric)
- Criteria for choosing an Azure Compute Service
Also know that when looking for alternatives, it's not uncommon to make use of multiple compute or other Azure service options by breaking up an older solution into parts at the time of such migration (for e.g. A serverless compute option like Azure Function + Service Fabric + something else if needed).
Generally speaking (and without knowing much about your application from your question currently), Azure App Service and Service Fabric could be considerations IMHO when migrating from an existing Cloud Service, but this is exactly where detailed requirements help you in decision making.
On a side note, here is a list of Azure Services available in CSP - Available Azure services in Azure CSP