Trespassers will be prosecuted

Simple trespass is not an offence. You can't be prosecuted for it. The signs that say you will be are, basically, lying.

Trespass combined with other factors can be an offence - for example, it's an offence to trespass in a way that is likely to cause fear in another person - and if any of those factors are present you can be prosecuted. Otherwise the landowner's only court option is to sue you in the civil courts for damages (for which he will need to show that he has suffered damage) and, if your trespass is ongoing, for an order for ejectment, to get you out.

I think your redress is to sue for any damage done. Otherwise you can merely ask them to leave.

Trespass on Railways and possibly other places is against the Law I believe.

In the broadest sense, yes.

If the interlopers are causing a disturbance to you, putting you at unease or denying you the right to peaceful enjoyment of your property (assuming they're not causing any damage), then you could bring a civil trespass case against them. Civil trespass is usually reserved for long-running disputes though, not single incidences of someone on your land.

Realistically if you contact the Gardai they can instruct the person to move on in the interests of public order. If they refuse to do so the Garda can then arrest them for a public order offence.

Farmers get very uppity about this sort of stuff.

There's no Right to Roam in Ireland, unlike the UK, but there's nothing a farmer can do except tell you to leave.

The Gardai will only come if you refuse to leave and become a nuisance, but then the Gardai will just tell you to leave too.

All land in Ireland is owned by somebody, even if in some cases "somebody" is the state or a state agency. Even if land is in state ownership taht doesn't necessarily mean that you have a right to enter it.

If you inadvertently trespass on somebody's lad by crossing an unmarked boundary, the fact that the boundary is unmarked tells you that the owner is not that bothered about trespassers, so the chances of any action being taken against you are low. Trespass generally isn't a problem on wilderness land, because there is no risk of harm resulting from gates left open, fences damaged, and the like, so owners are mostly unbothered.

If you have a dog there's a chance it'll get lead poisoning if you don't keep it on it's lead if you are on someone's land.

With good reason. Do you see the state people leave places after them on their 'wild' camping expeditions? Setting fires, leaving rubbish, breaking gates and fences.

Ramblers with dogs when faced with signs reading NO DOGS, ah shur my liddle schnookums wouldn't harm a fly. Signs with 'All dogs on leads' are heartily ignored.

In scandi countries there is a right to roam, but of course they behave themselves unlike the shower here.

Sure. I'm not disputing the rights of farmers to shoot uncontrolled dogs, or criticising those who do.

It's just that I've done a fair amount of walking in upland grazing areas in Wicklow, Down, Kerry and other parts of the country. I've seen a lot of sheep, but not so many attending farmers. I'm not suggesting that farmers are neglecting their livestock, but checking on them from time to time is not the same things as being in constant attendance.

Mind you, I never brought a dog up, and I saw very few dogs being brought up by anyone else. Perhaps if their had been more dogs around shotgun-bearing farmers would have appeared from out of the peat hags and behind the furze bushes.

I realise there are many places in the countryside that seem to be wilderness but are actually owned by farmers or the state etc but didn't realise private gardens without walls or hedges are the same, eg Bayside estate in Dublin is one example where most of the front gardens in that estate are open, so theoretically anyone could sit in those front gardens and have a picnic, sunbathe etc and they wouldn't be trespassing. I'd say a lot of people would be as surprised as myself to hear that.

There's no general right to enter unfenced land.

The point about land being unfenced is that it indicates that the landowner isn't unduly bothered about the risk of livestock straying, or about the possibility of people entering - if he weres, he'd fence it. So you might that that, if you do enter that land, no-one is likely to object. But this doesn't mean that you have a right to enter, or that if you do enter you are not trespassing or can't be asked to leave.