STLink v2 clone; connected via SWD using the stlink util
Are you using the st-link programmer software? This has an option "connect under reset" somewhere. Try setting this option, while pressing reset click on "connect", release reset button. Helped me before.
Good catch. It could be a problem such as this. But for this option to work, you have to connect the reset pin to the matching pin of the jtag adapter (NRST?)
If you can't do that, one trick to try is to hold the µC in reset manually just before attempting to connect to it and then release the reset.
The issue there if it solves the connection problem would most likely be that the µC internal clock is stopped. When it is, you can't connect JTAG or SWD. This is an issue if your µC for instance is in stop mode while you're attempting to connect to it.
In the same vein, does the STM32F101 has an internal oscillator that it can start from? I have never used the STM32F1 family (only STM32L4 and F4, that do have an internal oscillator). If suppose it does, but if not, you may need to wire an external crystal or such.
All in all, if you're sure about power supply and SWD connections, it may be this kind of oscillator problem - something to check. SWD will not work without the µC clock running.
You could also try to lower the SWD frequency in the settings.