In DOS, if you want graphics above the VGA-modes, you need the VESA BIOS EXTENSIONS (VBE). Often cards only support a limited subset of VESA modes and there are different versions of the VESA standard (1.2, 2.0, 3.0).
You can gain full VESA support if you have a card specific VESA driver.

FreeBE/AF

FreeBE/AF is a free software project to support full VESA and VBE/AF (2D VESA graphics acceleration) on a number of cards.
It contains drivers for ATI, Avance Logic, Cirrus, Matrox, Nvidia, Paradise, S3, Trident, Tseng, V7.
FreeBE/AF homepage

UNIVBE

There was a freeware tool called UNIVESA and later UNIVBE that gave VESA support to a set of graphics hardware. It was later part of the shareware Scitech Display Doctor, which is no longer developed.
Download of Scitech Display Doctor
Old freeware UNIVBE

ATI

ATI provided 64vbe for some legacy cards like mach64 and rage128. They no longer provide it for download, latest version is v2.21.
64vbe v2.21
FTP-mirror with 64vbe

Matrox

There is a driver written from Carsten Sorensen for the G100, G200 and G400-chip. It's called Gx00VBE and works very well.
Gx00VBE homepage

Other/Untestet

SiSVBE v0.83 (4K)
Trident VESA (8K)

Collection

Collection of VESA-Drivers
I found this one while searching for several VESA-drivers. It contains several drivers for older cards. Try it if you don't find any other help. Also contains the last free Version of UNIVBE.
Contains drivers for: AHEAD, APPIAN, ATI, BOCA, CIRRUS, COMPAQ, C_AND_T, DIAMOND, EVEREX, GENOA, HEADLAND, HERCULES, IBM, IRIS, OAK, ORCHID, PARADISE, PERFECTV, SIGMA, SPIDER, STB, TECMAR, TI_PORT, TRIDENT, TSENG, UNIVESA, VIDEO7, WESTERN

Tools

VINFO will display your gfx board's VBE version and all supported videomodes.
It's written by StyX/HeadcrasH.
Download VINFO