This is what I see:

1>------ Build started: Project: xxx (xxx\xxx), Configuration: Debug Win32 ------
1>C:\Program Files (x86)\MSBuild\Microsoft.Cpp\v4.0\V120\Microsoft.Cpp.Platform.targets(64,5): error MSB4018: The "VCMessage" task failed unexpectedly.
1>C:\Program Files (x86)\MSBuild\Microsoft.Cpp\v4.0\V120\Microsoft.Cpp.Platform.targets(64,5): error MSB4018: System.FormatException: Index (zero based) must be greater than or equal to zero and less than the size of the argument list.
1>C:\Program Files (x86)\MSBuild\Microsoft.Cpp\v4.0\V120\Microsoft.Cpp.Platform.targets(64,5): error MSB4018:    at System.Text.StringBuilder.AppendFormat(IFormatProvider provider, String format, Object[] args)
1>C:\Program Files (x86)\MSBuild\Microsoft.Cpp\v4.0\V120\Microsoft.Cpp.Platform.targets(64,5): error MSB4018:    at System.String.Format(IFormatProvider provider, String format, Object[] args)
1>C:\Program Files (x86)\MSBuild\Microsoft.Cpp\v4.0\V120\Microsoft.Cpp.Platform.targets(64,5): error MSB4018:    at Microsoft.Build.Shared.ResourceUtilities.FormatString(String unformatted, Object[] args)
1>C:\Program Files (x86)\MSBuild\Microsoft.Cpp\v4.0\V120\Microsoft.Cpp.Platform.targets(64,5): error MSB4018:    at Microsoft.Build.Utilities.TaskLoggingHelper.FormatString(String unformatted, Object[] args)
1>C:\Program Files (x86)\MSBuild\Microsoft.Cpp\v4.0\V120\Microsoft.Cpp.Platform.targets(64,5): error MSB4018:    at Microsoft.Build.Utilities.TaskLoggingHelper.FormatResourceString(String resourceName, Object[] args)
1>C:\Program Files (x86)\MSBuild\Microsoft.Cpp\v4.0\V120\Microsoft.Cpp.Platform.targets(64,5): error MSB4018:    at Microsoft.Build.Utilities.TaskLoggingHelper.LogErrorWithCodeFromResources(String messageResourceName, Object[] messageArgs)
1>C:\Program Files (x86)\MSBuild\Microsoft.Cpp\v4.0\V120\Microsoft.Cpp.Platform.targets(64,5): error MSB4018:    at Microsoft.Build.CPPTasks.VCMessage.Execute()
1>C:\Program Files (x86)\MSBuild\Microsoft.Cpp\v4.0\V120\Microsoft.Cpp.Platform.targets(64,5): error MSB4018:    at Microsoft.Build.BackEnd.TaskExecutionHost.Microsoft.Build.BackEnd.ITaskExecutionHost.Execute()
1>C:\Program Files (x86)\MSBuild\Microsoft.Cpp\v4.0\V120\Microsoft.Cpp.Platform.targets(64,5): error MSB4018:    at Microsoft.Build.BackEnd.TaskBuilder.<ExecuteInstantiatedTask>d__20.MoveNext()
========== Build: 0 succeeded, 1 failed, 0 up-to-date, 0 skipped ==========

and the solutions listed in the google and SO searches don't deliver (removing trailing slashes, reverting to default $(OutDir) or $(TargetDir) paths, etc.)

Context

This regards several projects which have been migrated from MSVC2012. Other projects with the same project/solution layout migrated from the same MSVC2012 environment didn't blink an eye, while some exhibit this error, which prevents any sort of successful compile/build result.

The Fix

The error was due to the project configuration field Platform Toolset being empty. (The .vcproj project file should list a value such as v120 here)

Before:

<PlatformToolset>
</PlatformToolset>

After:

<PlatformToolset>v120</PlatformToolset>

This value can be set in DevStudio via project Properties panel > Configuration Properties > General > Platform Toolset :: pick one of the values from the dropdown there.

The way this was uncovered:

create an empty new project in MSVC2013 of the same type
use Beyond Compare (or another visual diff tool which can easily edit the files being compared by copying chunks across) to copy the files set across and test (by reloading into DevStudio and running the build process in there).
next copy across bits of the project settings (and duplicate the blocks if your original project has multiple build targets like mine had (Win32/x64/Itanium)), test again.
rinse and repeat the above until your 'new' project exhibits the same issue as the old one. Narrow down to a minimal XML subset in the .vcproj file.