I'm assuming the desktop compiler works in Vista (32 bit Home Premium).
It does, I have both XP and Vista machines and it works on either. Basic4ppc is built on the .NET Framework. There are two compilation modes, "legacy and optimised".
Legacy mode will run on devices with .NET Compact Framework 1.0 SP3 or later and desktops with .NET Framework 1.1. This mode is byte code interpreted.
Optimised mode requires .NET Compact Framework 2.0 or later on the device and .NET Framework 2.0 or later on the desktop. This mode is fully compiled and has much higher performance. It uses the .NET C# compiler for the final code production.
Unless you are using a pirated version of Basic4ppc you won't be able to try this as the compilation facilities are disabled in the trial version. You should still be able to run programs on your device and desktop using the respective IDEs and compile them when you purchase the full version.