What GPU cards will run Windows 10 64bit?    On Desktops only ?     

Warning #1  Do not buy $100+ video cards  from CHINA, sold for $25, those are fake cards.  My page has zero to do with BITCOIN use of ANY KIND.
If you upgraded say VISTA PC from year 2007 and find your self upgrading to Windows 10, the iGPU will mostly fail in the CORE2 CPU. The cure is simple buy a used $10 GPU card and instal it per the below.
GPU card upgrades are complex!    (12 makers) and huge sets of features, ports,  speeds, Directx12 support and DRM and textures and cores. 
If the  W10  loaded PC shows in device manager  under DISPLAY ADAPTERS as  MBDA  driver, this a limphome crippleware
driver, from Microsoft (MS) but is very cool with screen not dead sure.
Intel HD iGPU or GMA internal GPU  rules.
If you have an  old PC most  old Intel HD chips (iGPU) have no w10 support for HD3000 and older (1 exception 2500) see the truth here.
There are like 10,000 cards made in last 15 years and many older ones do not work now in W10. (means full support, not crippled drivers like MBDA is...)
First lets do NVIDIA GEFORCE GPU,  The Series 7 up are supported but not series 6 and below.
NVIDIA.com  (Geforce  series 7 and up for W10 support   (GT705  is bottom tier series 700)
There are some old cards that work. 
See my cheap list below:
I have a Series  8400GS card that turns 5-FPS,   a $5 used  card and pathetically slow,  if not gaming this works. It is cheap 2D card.
Next up?
Just as good is  AMD.com (ATI) Radeon™ etc..  Their list here and is SUPER GOOD , just one page, all you need to know buying used AMD cards, and wanting W10 support.

The winner for low cost AMD GPU card that runs W10 and directx-11,  is HD 5450 (DVI , avoid DMS59 version)  at the eye popping cost of $8
This  5450 does a whopping  5-FPS, (frame per second) it is a slug, but again is only $8 card.  This is only a starting point, so mow 10 lawns and upgrade to a faster card  in this list "super good".
Many chips made for cards, long ago ended support for their chip. (if true the card maker can not help you)

That is my answer simple and sweet, which one goes fast enough for you is any ones guess. (to get high FPS gaming? )
High end:
But the ratings on all cards are HERE . I have a  GTX1650 and love mine, (runs top games too, AAA+)

Buy what you can afford and be happy gaming?  (GPU upgrades never end for true gamers)

The card must fit the PCI-express slot,  and not hit things inside (dah factor 1 FITS not HIT),  and match low profile case or high-profile cases.
It must not overload the PSU nor cause small cases to overheat.(added case fans will allow faster GPU gaming cards)
The top gaming cards have 2 PCIE, PSU aux cables, to the GPU cards rear power jacks  6 pins or 8 or mix.  (look first then leap) those jacks and PSU cables match are NOT optional when seen.

PCs that fail to load W10 and not fail are caused by PCs older than year 2007 made (I have 2005 laptop that runs W10-32bit but not 64)
Any  PC older than CORE2 Q41 chipset seen here, will fail for iGPU zero intel support, but  adding  8400GS card cures that. for $10-$15 bucks.

DirectX support page for top CARDs
Nvidia shows GT730 lowest card with Directx 12.

12 makers of cards. (GPU) PRIME not junk not clones , counterfeits from China. The Top 7 are best.
  • NVIDIA (chip and card both) (#1 maker  here}
  • AMD/ old ATI (chip and card both)"RADEON"
  • ASUS (and some of the finest DT MOBO on earth and GPU too.) {AMD&Nvidia}  Top brand here.!
  • GIGABYTE (both AMD and Nvidia)  Big time player and top quality and support , good specs  and documents and forum , a 4 star maker.
  • Zotac  (one of my favorites , Nvidia focused)
  • PNY (Nvidia focused)
  • EVGA (  Nvidia chip sets.) {spec forgets card watts, -1 point, no specific manual on 1 exact card -5 point. sure, how to use water cooling +1, but no real manual for all cards sold even now} (I PASS)
  • Sapphire (IDK)
  • XFX (Pine?) (IDK)
  • VisionTek? (IDK)
  • Powercolor (not me, ever)
  • MSI ( has problems and sells (AMD & Nvidia)  I do not like MSI at all. ( their documents STINK) IMO.

(the w10 cheap list by me) Nvidia fan me. (AMD is ok) as cheap used as $10 no w10 errors and are office grade 2D use and light photoshop, never are this any kind of Gamers card.
GeForce 400 Series:
GeForce 405
GeForce 300 Series:

GeForce GT 340, GeForce GT 330, GeForce GT 320, GeForce 315, GeForce 310
GeForce 200 Series:

GeForce GTX 295, GeForce GTX 285, GeForce GTX 280, GeForce GTX 275, GeForce GTX 260, GeForce GTS 250, GeForce GTS 240, GeForce GT 230, GeForce GT 240, GeForce GT 220, GeForce G210, GeForce 210, GeForce 205
GeForce 100 Series:

GeForce GT 140, GeForce GT 130, GeForce GT 120, GeForce G100
GeForce 9 Series:

GeForce 9800 GX2, GeForce 9800 GTX/GTX+, GeForce 9800 GT, GeForce 9600 GT, GeForce 9600 GSO, GeForce 9600 GSO 512, GeForce 9600 GS, GeForce 9500 GT, GeForce 9500 GS, GeForce 9400 GT, GeForce 9400, GeForce 9300 GS, GeForce 9300 GE, GeForce 9300 SE, GeForce 9300, GeForce 9200, GeForce 9100
GeForce 8 Series:

GeForce 8800 Ultra, GeForce 8800 GTX, GeForce 8800 GTS 512, GeForce 8800 GTS, GeForce 8800 GT, GeForce 8800 GS, GeForce 8600 GTS, GeForce 8600 GT, GeForce 8600 GS, GeForce 8500 GT, GeForce 8400 GS, GeForce 8400 SE, GeForce 8400, GeForce 8300 GS, GeForce 8300, GeForce 8200, GeForce 8200, GeForce 8100

Legacy 101 rule, best post on MS forum here:

"Personally... Don't do it. (highlights by me in red)

Your machine is too old for Win10.

Unless you don't need your machine to perform as well as it

would with Win7 and proper dedicated Win7 drivers,  and don't mind having to deal with increasing issues over time.

We're talking about  13/14 yr old hardware from the XP/Vista era  here.

As time goes by and more Win10 version updates are released the lack of compatibility will only get worse. (<<< I SAY read this 2 times for effect)

Stick with the OS that suits the hardware.... (OEM told you)."

The most confusing thing at Intel.com is the support page states CORE(tm) support, but fail to say, that does NOT include CORE2 Duo or CORE2 QUAD. (very very poorly stated IMO)

It is true as time advances, Microsoft PNP catalogs, drop dead drivers, dead means chip maker ended all support.

A cheap fix for W10 errors on iGPU is here  Nvidia 8400GS (about $20) note 3 jack types !! Even run HDMI port to any TV.

version 3.  1-20-2020 (The above is just my opinion )