I have installed many versions of ubuntu and mint and always had no difficulty. My problem this time is I've bought a new laptop with a tiny hard drive (30 Gb) windows 10 takes about 28 at present. Is it possible to install Linux on an external hard drive and have grub somehow work. Any advice or sugestions gratefully recieved

All I gotta say for starters. If wanting Windows untouched.

Install grub bootloader to mbr of external drive and get bios to control boot order when choosing what to run.

Install is just normal install but one must make sure the external drive is picked as the destination device.

Edit: when I say bios. Usually. There iskey menu option for just boot device order like hard drive, dvd drive, usb drive, etc....

https://forums.linuxmint.com/viewtopic.php?t=266484

Yes, basically just a normal install. That link seems a bit convoluted for this situation. Win10 most likely means UEFI - that means the Mint entry will be added to the EFI partition on the internal "disk" (probably actually an eMMC chip). It's only a small amount of data, but most likely will become the default entry - so you always need to have the external plugged to boot successfully. Easy to fix from Mint itself, or go into the firmware menus and change the default back to Win10.
Let us know how you go, and we can advise further.

I remove power/data to internal drive then install to the external just as if it were a normal hard drive.

For most modern linux, a usb will be as a hard drive. Older and special distro's may not. Consider the newer big name distro's that have full support for uefi secure boot or you will have to disable that every time.

Good advise. Like the OP I had the same problem. That feature doesn't exist/work in Mint/Ubuntu. On the Manjaro installer it work absolutely prefect.

If it's an eMMC as I suspect, it'll be soldered onto the m/b. No chance the OP can disconnect power.

after reading all the responses and thinking about it.
I can only free up 0.5Gb and would like my machine to boot without the external drive attached.
can I install over the internal drive and usb one. say putting /boot /sbin an /bin on the internal drive and having /var /home /usr /proc and the swap partition on the external drive.

will this work?

You might read the link below on dual booting windows 10 uefi with Ubuntu. If your windows 10 is EFI, Ubuntu will install its EFI files to the EFI partiton on the first drive and you can install the Ubuntu system on the external. If you don't plan to use Ubuntu/Mint that often, just leave windows as the default boot in your firmware.