I have a number of local windows, Linux and macs on my local LAN, but no matter what, Safari will not resolve them by name.
How can I get this to work as there is no "server" on my network. I use the machine names only without suffixes.
It's likely that the computers are using NetBIOS name resolution (aka WINS or occasionally SMB). This is used commonly for Windows file sharing, and is typically set when you enter a hostname for a computer. On Macs and Linux, it's not always broadcast, unless Windows file sharing is enabled/installed, but it typically is, so it's usually a workable option for small networks without a lot of fuss.
However, iOS devices don't use NetBIOS (since they don't do Windows file sharing, there's not a lot of point, and there may be licensing issues, I'm not sure). So unless you've explicitly set up DNS hostnames on your network, this is most likely why you can't address your machines by name from your iPad.
The best solution here is to assign hostnames using static DHCP on your router — because your machines most likely already use your router for DNS, you should be able to change your settings on the router without needing any changes on any of the machines. Most home routers have some way of doing this, but it varies from model to model. The basic process is to assign a DHCP reservation for each machine and add the host name (some routers might allow for a hostname based on MAC address, without a static reservation). Those host names should then be resolvable from all machines, including the iPad via DNS.
If you need further help with this, add some information about your router (although searching for an answer on Superuser might yield the information you need).