How to get a list of all routes in ASP.NET Core?

To get at all the routes, you need to use the ApiExplorer part of MVC. You can either mark all your actions with an attribute or use a convention like this one:

public class ApiExplorerVisibilityEnabledConvention : IApplicationModelConvention
{
    public void Apply(ApplicationModel application)
    {
        foreach (var controller in application.Controllers)
        {
            if (controller.ApiExplorer.IsVisible == null)
            {
                controller.ApiExplorer.IsVisible = true;
                controller.ApiExplorer.GroupName = controller.ControllerName;
            }
        }
    }
}

In Startup.cs, add your new in ConfigureServices(...)

public void ConfigureServices(IServiceCollection services)
{
    services.AddMvc(
        options => 
        {
            options.Conventions.Add(new ApiExplorerVisibilityEnabledConvention());
            options.
        }
}

In your ActionFilter you can then use constructor injection to get the ApiExplorer:

public class MyFilter : ActionFilterAttribute
{      
    private readonly IApiDescriptionGroupCollectionProvider descriptionProvider;

    public MyFilter(IApiDescriptionGroupCollectionProvider descriptionProvider) 
    {
        this.descriptionProvider = descriptionProvider;
    }

    public override void OnActionExecuting(ActionExecutingContext actionContext)
    {
        base.OnActionExecuting(actionContext);

        // The convention groups all actions for a controller into a description group
        var actionGroups = descriptionProvider.ApiDescriptionGroups.Items;

        // All the actions in the controller are given by
        var apiDescription = actionGroup.First().Items.First();

        // A route template for this action is
        var routeTemplate = apiDescription.RelativePath
    }
}

ApiDescription, which has the RelativePath, which is the route template for that route:

// Copyright (c) .NET Foundation. All rights reserved.
// Licensed under the Apache License, Version 2.0. See License.txt in the project root for license information.

using System;
using System.Collections.Generic;
using Microsoft.AspNetCore.Mvc.Abstractions;
using Microsoft.AspNetCore.Mvc.ModelBinding;

namespace Microsoft.AspNetCore.Mvc.ApiExplorer
{
    public class ApiDescription
    {
        public string GroupName { get; set; }
        public string HttpMethod { get; set; }
        public IList<ApiParameterDescription> ParameterDescriptions { get; } = new List<ApiParameterDescription>();
        public IDictionary<object, object> Properties { get; } = new Dictionary<object, object>();
        public string RelativePath { get; set; }
        public ModelMetadata ResponseModelMetadata { get; set; }
        public Type ResponseType { get; set; }
        public IList<ApiRequestFormat> SupportedRequestFormats { get; } = new List<ApiRequestFormat>();
        public IList<ApiResponseFormat> SupportedResponseFormats { get; } = new List<ApiResponseFormat>();
    }
}