PPA (Personal Package Archive) are used to include a specific software to your Ubuntu, Kubuntu or any other PPA compatible distro. The "safeness" of a PPA depends mostly on 3 things:
Who made the PPA - An official PPA from WINE or LibreOffice like ppa:libreoffice/ppa and a PPA that I created myself are not the same. You do not know me as a PPA maintainer, so the trust issue and safety is VERY low for me (Since I could have made a corrupted package, incompatible package or anything else bad), but for LibreOffice and the PPA they offer in their website, THAT gives a certain safety net to it. So depending on who made the PPA, how long he or she has been making and maintaining the PPA will influence a little bit on how safe the PPA is for you. PPA's as mentioned above in the comments are not certified by Canonical.
How many users have used the PPA - For example, I have a PPA from http://winehq.org in my personal PPA. Would you trust ME with 10 users that confirm using my PPA having 6 of them saying it sucks than to the one Scott Ritchie offers as ppa:ubuntu-wine/ppa in the official winehq website. It has thousands of users (including me) that use his PPA and trust his work. This is work that has several years behind it.
How updated the PPA is - Let us say you are using Ubuntu 10.04 or 10.10, and you want to use THAT special PPA. You find out that the last update to that PPA was 20 years ago.. O.o. The chances you have on using THAT PPA are null. Why?. Because the package dependencies that PPA needs are very old and maybe the updated ones change so much code that they wont work with the PPA and possibly break your system if you install any of the packages of that PPA to your system.
How updated a PPA influences the decision to use it if he/she wants to use THAT PPA. If not they would rather go look for another one more up to date. You do not want Banshee 0.1 or Wine 0.0.0.1 or OpenOffice 0.1 Beta Alpha Omega Thundercat Edition with the latest Ubuntu. What you want is a PPA that is updated to your current Ubuntu. Remember that a PPA mentions for what Ubuntu version is made for or multiple Ubuntu versions was made for.
As an example of this here is an image of the versions that are supported in the Wine PPA:
Here you can see that this PPA is supported since Dinosaurs.
One BAD thing about how updated a PPA is, if the PPA maintainer tends to push into the PPA the latest, greatest and cutting edge version of a specific package. The down side of this is that if you are going to test the latest of something, you ARE going to find some bugs. Try to stick with PPAs that are updated to a stable version and not a unstable, testing or dev version since it might/will contain bugs. The idea of having the latest is also to TEST and say what problems were found and solve them. An example of this are the daily Xorg PPAs and Daily Mozilla PPAs. You will get about 3 daily updates for X.org or Firefox if you get the dailies. This is because of the work the put in there and if you are using their daily PPAs it means you want to help with bug hunting or development and NOT for a production environment.
Basically stick with this 3 and you will be safe. Always look for the maker/maintainer of the PPA. Always see if many users have used it and always see how updated the PPA is. Places like OMGUbuntu, Phoronix, Slashdot, The H, WebUp8 and even here in AskUbuntu are good sources to find many users and articles talking about and recommending some PPAs that they have tested.
Stable PPA Examples - LibreOffice, OpenOffice, Banshee, Wine, Kubuntu, Ubuntu, Xubuntu, PlayDeb, GetDeb, VLC are good and safe PPAs from MY experience.
Semi Stable PPA - X-Swat PPA is a in the middle PPA between bleeding edge and stable.
Bleeding Edge PPA - Xorg-Edgers is a bleeding edge PPA although I should mention that after 12.04, this PPA has become more and more stable. I would still mark it as bleeding edge but it is stable enough for end users.
Selectable PPA - Handbrake offers here a way for the user to choose, do you want a stable version or do you want the bleeding edge (Also referred to as Snapshot) version. In this case you can select what you want to use.
Note that in the case of using for example the X-Swat ppa with the Xorg-Edgers PPA, you will get a mixed between the two (With priority towards Xorg-Edgers). This is because both are trying to include almost the same packages, so they will overwrite each other and only the most updated one will show in your repositories (Except if you manually tell it to grab the package from X-Swat).
Some PPAs might update some of your packages when you add them to your repository because they will overwrite with their own version a certain package to make the PPA software work on your system correctly. This might be some code packages, python versions, etc.. Other like the LibreOffice PPA will remove all existence of the OpenOffice from your system to install the LibreOffice packages there. Basically read what other users have commented about a specific package and also read if the package is compatible with your Ubuntu version.
As the comment below suggest by Jeremy Bicha, some bleeding edge (PPAs that stay very up to date including adding Alpha, Beta or RC quality software in the PPA) could potentially damage your whole system (In the worst case). Jeremy mentions an example of many.
Best Answer
Snaps have the https://snapcraft.io/ repo. This is run by Canonical, the same people that build Ubuntu.
Flatpaks have an official repo at https://flathub.org/ . Flatpaks were developed by Redhat but I don't know if they manage the flathub repo or not.
Stability
The stability of the individual packages, of course, rely on the quality of the build and are at the mercy of the maintainer.
Both flatpaks and snaps are built completely using the dependencies they need inside a sandbox but both handle this a little differently
Snaps build a mount point and the system mounts the program archive and runs it from there.
Flatpaks are built in
/var/flatpak/for system-wide (global) installs and in~/.var/appon the local side. It mounts those and runs them.The good news about stability is that if you get a wonky application in either it is contained and will not make the rest of your system unstable by installing libraries that cause conflicts with other installed apps.
Both are self-contained applications with all the needed information to run. This is what makes this distro-agnostic and allows them to be installed on any Linux system that supports them (flatpak or snap)
Security
This is a little more ambiguous.
Snaps only have the official repo. There was one reported case of malware getting into the repo but it was caught quickly and removed. It was cryptocurrency mining software that would send some mined currency back to the app maintainers without the users knowledge. Even with that there was no other ill effect from the app and AFAIK, it was unable to access the home folder of the user.
Flatpaks: If you use the official repo it should have about the same security as Snaps, nothing is perfect but anything that makes it in will be very quickly noticed and removed if it is malware and made it past the initial submission review.
I would personally doubt that anything overtly malware like a virus would make it into either Snap's or Flatpak's repos and anything with sneaky unwanted behavior like the aforementioned cryptocurrency mining app would stay in very long.
Overall I would say that both are safe but neither is as inherently safe as the official Ubuntu sources, but this goes for PPAs as well. Adding any sources outside of Ubuntu's official sources is not quite as safe.
I do have to add a caveat here, there are other Flatpak repos out there. Most of these are for legitimate programs that just want to host their own repo rather than use flathub. Those are completely outside any quality control of flathub and should only be added if you trust the developers of the program. This would also go for adding snap repos but I don't think that at this time there are any but the official Snap repos.
As to the whether or not flatpaks and snaps are safe to install
Overall that are safe as long as you stick to the official repos, look over the description of the packages you want to install and don't install anything that looks even a bit shady.
Both are a great way for users to have a safe (as safe as can be expected outside a distro's official package sources) way to install software that are not available any other way and have them "just work".
For example, I have Spotify installed as a Snap and Teamspeak 3 installed as a flatpak. While Spotify is available via a ppa, using a snap allows me to avoid cluttering apt with PPA that I can avoid using.
Teamspeak would only be available for me with a .run that unpacks the folder and then you put the extracted folder in your home directory and click on the sh file or use the command line to start. While I did this before and then made a desktop launcher to launch it for me then added that launcher to my
~/local/share/applicationsfolder to launch it. It was so much easier to just install the Flatpak in one step and have it work.To address that part of your lead up to your question:
I would suspect that reason the PPA completely hosed your Ubuntu install is because it brought in newer libraries as dependencies that your native programs were unable to use or overwrote your installed libraries with older ones that were too outdated to be used by your native Ubuntu.
The good thing about both snaps and Flatpaks is that they will bring in any libraries they need to run inside their own folders. Snaps and Flatpaks are self-contained and will not touch any of your system files or libraries.
The disadvantage to this is that the programs might be bigger than a non snap or Flatpak version but the trade off is that you don't have to worry about it affecting anything else, not even other snaps or Flatpak. If the app is broken because it brought in bad libraries or for any other reason you just uninstall it and it is completely gone.