This is just an introduction, primarily for those new to Linux in general, Puppy in particular. Following up a couple of recent posts revealed that there have been discussions scattered about the forum, but no specific thread from which newbies can get their bearings.
Feel free to flesh out this thread, adding recommendations and links.
The simplest tool --for the simplest projects-- is probably avidemux. There are likely versions which will run under any Puppy and Windows programs which will run under Wine. If all you want to do is cut and trim then combine the results, avidemux will do the job. It can also manage simple clip rotations, and codec conversions. If you're familiar with VirtualDub under Windows, avidemux offers comparable capabilities. [Absent 1/2 : Under Windows there was a companion program to VirtualDub named Cut-Assistant. Calling VirtualDub via Cut-Assistant enabled you to create a 'play-list' of 'cut-clips' to be combined into one output file; or, alternatively, the output file would consist of everything EXCEPT what you 'cut-out'. Although VirtualDub can be run under Wine, Cut-Assistant can't].
Similar to avidemux --but only for 64-bit operating systems-- are Lossless-Cut and VidCutter.
At the other extreme is Cinelerra. There is likely some version which will run under any Puppy. And AFAIK there isn't anything Linux Video-Editors can do which Cinelerra can't. But it's down-side is that it's GUI isn't very intuitive. You may face a significant learning curve. On the other hand, if you aren't already accustomed to some 'full-fledged' Video-Editor, you won't have much to 'unlearn'.
What prompted this post was don570's recent report of the utility of the Shotcut AppImage on the FatDog64 SubForum. Shotcut was first introduced to us as a builtin under FatDog64, I believe FatDog64-630. Now only published as 64-bit programs and applications, there are older 32-bit versions. I can't tell you anything about how functional those are. Windows version 15.05.02 will run under Wine. (Haven't tried newer versions).
Shotcut appears to be a solid, all-purpose video-editor with an easy to understand GUI. But for a near professional quality video-editor [other than Cinelerra] you'll probably want either Openshot or KDEnlive.
There were some 32-bit KDEnlive versions for Slackos & Saluki/Carolinas. But until Bionicpup Openshot 1.4.x was generally the 'go-to' full-fledged video-editor. Openshot 1.4.3 ran well under Xenialpup, both 64 and 32 bit versions. But with the qt5 and/or python changes which went into the versions published for Ubuntu Bionic Beaver, KDEnlive resulted in a better application for Bionicpup. As far as I can tell, their GUIs are almost identical. But, if you are a confirmed Openshot user, or have projects which would be difficult to work with under KDEnlive, I suggest you look for Mike Walsh's Openshot 1.4.3.sfs. [The designation "2" indicates the start of the series using Qt5].
Caution: Openshot and KDEnlive conflict with each other. If you want both, it is recommended that they be run as SFSes --with only one loaded at a time-- or as self-contained AppImages.
IIRC, you can simultaneously run avidemux, shotcut and KDEnlive.