ASM1042 is often paired with ASM1051E as a chipset in order to provide 2 USB 3.0 ports as well as one SATA 6Gbps port on the same card. ASM1051E does not show up in lspci even though it is there, working. Works with gNewSense 3.0 LiveDVD and Trisquel 6.0 LiveCD out-of-the-box. I don't have any USB 3.0 devices to test with it, but USB 2.0 devices seem to work fine. SATA HDD drives can also be detected while SSDs cannot (August 2013).

lspci -vmmnn

Slot:   04:00.0
Class:  USB controller [0c03]
Vendor: ASMedia Technology Inc. [1b21]
Device: ASM1042 SuperSpeed USB Host Controller [1042]
SVendor:        ASUSTeK Computer Inc. [1043]
SDevice:        Device [8488]
ProgIf: 30

lspci -nnk

04:00.0 USB controller [0c03]: ASMedia Technology Inc. ASM1042 SuperSpeed USB Host Controller [1b21:1042]
        Subsystem: ASUSTeK Computer Inc. Device [1043:8488]
        Kernel driver in use: xhci_hcd
        Kernel modules: xhci_hcd

Another system using the controller on a mini PCIe card. (February 2020)

sudo lspci -v

02:00.0 USB controller: ASMedia Technology Inc. ASM1042 SuperSpeed USB Host Controller (prog-if 30 [XHCI])
        Subsystem: ASMedia Technology Inc. ASM1042 SuperSpeed USB Host Controller
        Flags: bus master, fast devsel, latency 0, IRQ 17
        Memory at fea00000 (64-bit, non-prefetchable) [size=32K]
        Capabilities: [50] MSI: Enable- Count=1/8 Maskable- 64bit+
        Capabilities: [68] MSI-X: Enable+ Count=8 Masked-
        Capabilities: [78] Power Management version 3
        Capabilities: [80] Express Legacy Endpoint, MSI 00
        Capabilities: [100] Virtual Channel
        Kernel driver in use: xhci_hcd

Note that in this case, the mini PCIe card also required power to be supplied to it via a four pin connector that used an auxiliary power cable to share power with SATA hard drives.