I have Linux installed on my notebook, see what it looks like.

Running Logger 32 in Linux, see the screenshot.

You will need genuine Windows module "dcom98", the easiest way to install it is to use the "winetricks" script:

install the winetricks:
wget http://www.kegel.com/wine/winetricks

Also, some winetricks "packages" require the cabextract tool to be installed. The cabextract tool is for extracting Microsoft cabinet files, also called .CAB files. Linux users can usually get it via their distribution's package management system, or visit http://www.cabextract.org.uk/ to download.

make winetricks executable:
chmod +x winetricks

run it:

sh winetricks

Program window opens, tick the "dcom98" option and install it.

Now install Logger32 as you install Windows applications under Wine, it should work like a charm :)

In case you use use USB interfaces for CAT/CW/PTT like Microham or Rigexpert, you might need to create symlinks to make your  system see the "comports"

ln -s /dev/ttyUSB0 ~/.wine/dosdevices/com1
ln -s /dev/ttyUSB1 ~/.wine/dosdevices/com2

Same about N1MM contest logger except you have to install mfc42.dll just like you install dcom98 in case of Logger32.