Mic through guitar amp

you CAN do it with any F-XLR to 1/4 Jack cable, but realistically it's going to sound pretty crap. Guitar amps are designed to sound good with guitars, not replicate the human voice. The same reason a guitar through a PA with no speaker simulator tends to sound horrible. A little portable PA system or an amp that's designed to be used like that (Roland Street Cubes are ideal) aren't that much more expensive and will give you much better results. For a good, inexpensive vocal mic, I really like the sE V7.

Look at bass "amps" some of which are just glorified PAs and can be more versatile than a regular PA speaker

This article explains it well. Tl;Dr - it probably works but the quality won't be great. Be careful with feedback by coming to close to the amp with the mic. Also a guitar amp usually stands behind you, which opens up more possibility for feedback.

It probably will work but it's only a temporary solution. A solid mic for under 100€ is the Audio Technica AT2020.

I've never done this so I'm not sure, but here are some thoughts. If you want to use it for vocals and the guitar at the same time, you'll need an amp with a separate aux input, something like the Fender Mustang.

Mics often require phantom power and a preamp, and typically they have an XLR connector. Some guitar amps have XLR input, but I don't think it's that common. Using the mustang as an example again, the aux input is 1/8 inch You'll probably need a preamp with 1/8 inch or 1/4 inch outputs or a preampe with XLR output and an adaptor. This thing looks like it would work, but double check and adjust to your needs: https://www.ebay.com/itm/394675855391?hash=item5be483a01f:g:rIQAAOSwRQNkh0TU&amdata=enc%3AAQAIAAAA8BxcuD4xgd1reK3dnW%2Bk3BPNxKcXp06ZNoX2j3lwqx5nJboOso9lzio8vY65wyraZXTPtmFaf3rvP854syN%2FOiWcXaRU%2FkdYSParo7ZuceF1ex%2FMxafeQFX8MaQBcTkychsIchVuJSOii2PfLhtHABu6lRaRB6WhDKzleuF1KPwHxW3z%2BfuuPVub8XsIzd7brLmQs%2F41LUO7wsBNVgWVvFVtfGpUZ2KxLmOge%2FiiNfuykqGjtV8D14H9EvQYXyP0yh1YvdmlqtLvEFSlbkmgE4YS%2FscNAV2Joy7SKF%2FiOdKa04XsLRluPhA6k4RD9fJCUg%3D%3D|tkp%3ABk9SR5zfzMeWYg.

I don't connect to a guitar amp, but I use the focusrite Scarlett 4i4 as an audio interface, and with adapters it might work. With something like that, you can also connect it to your computer for recording.

I've used an MXL 770, and for the price it sounds nice but you need to be close. It's more for talking than singing and I don't know if it's what you want.

You're going to need to consider feedback.

I'm no expert, so hopefully someone with more knowledge has better advice, but maybe this gets you somewhere.

Some acoustic focused amps would have a separate xlr input for a mic.

If you don't have one of those then you're pretty much out of luck. Mic and instrument level are different things, so your amp won't be set up for the mic signal level.

I'd go for a cheapo separate setup, either a dedicated mini PA or run the mic into a mixer and out through some speakers. Also be aware that some microphones require a powered connection to operate, although most all mixers will have at least one channel that can do this.

IMO the best bang for your buck mic is a Shure SM57. You can use it for everything from vocals to an amp to a snare drum.