Sega Master System / Mark III / Game Gear
This guide will assume you're using Meka.
Load the game you want to log music from. Click on the SOUND menu, then choose Dump, then VGM Start.
When you want to stop logging, choose VGM Stop from the same menu.
You should see messages like this in the Messages window, if you have it open:
The file will have been created in the \music\ subfolder of Meka's folder.
These are generally only needed when recording sampled audio (ie. voices). For music, frame-accurate (which is what non-sample-accurate VGMs are) gives (almost) exactly the same results, and the files are smaller. So always leave the sample-accurate option unchecked unless it is really needed.
You should record plenty of music, until you are sure you have heard the whole of any looped section, plus quite a bit more. For safety, record the looped section twice, plus a bit more.
You might want to use speed-up features to "fast-forward" as you log the music; in Meka you can press F2 to switch to manual frameskip mode, and use F4 to speed it up to 1/9 (press F2 again to go back to automatic frameskip mode).
Most games were designed for play on 60Hz machines, in Japan or the US. Some games were designed and only released in Europe, and their music is probably designed for playback at 50Hz. In Meka, the TV Type setting affects the playback speed of the VGMs; in Dega it's the PAL menu item. The general rule is, record at NTSC/60Hz unless you have good reason to record at 50Hz, since if necessary the plugin can be set to play back 60Hz files at 50Hz.
Use the emulator's hard-pause facility (F12 in Meka). This will stop emulation (and VGM logging), unlike when you use the game's software pause facility by pressing the emulated Pause button. This will make Meka's GUI as fast as possible. This also (sort of) gives you a keyboard shortcut for VGM logging.
It's a lot easier to record a VGM if you can control when the music starts.
Make sure there are no sound effects played, because they will be recorded as well. This can be tricky... so here are some pointers: