How does the old saying go? When life gives you lemons, make lemonade. This week’s lemons are shares, as the UK has chosen to leave this universe and move to a parallel plane of existence. Or you could be forgiven for thinking something as bad as that has happened judging by the mass panic we’ve seen.

The lemonade? Dividends!

Oil doesn’t care about the EU

Royal Dutch Shell (LSE: RDSB) shares have shrugged off the madness, and are actually up a bit since the EU vote. At least the markets are behaving rationally in this case, seeing the long-term value of oil companies as being totally independent of what trading bloc the UK is or isn’t in.

The real beauty of Shell is the dividends it offers and with the share price still being down around 25% over the past 12 months, we’re looking at forecast yields of around 6.5% this year and next. The big question, and one of the issues helping keep the share price lower than it might be, has been whether those forecasts would actually be met. This year’s payment would be only around 70% covered by predicted earnings, with the 2017 dividend just squeaking through to fully covered.

But with the Q1 dividend already announced and confidence in the cash payments recovering, Shell shares have picked up 50% since their low point in January — but there’s still time to lock in that high yield before it erodes further.

Insurance tumble

The insurance sector is one that could be hit by our pending EU exit and Legal & General (LSE: LGEN) shares have tumbled more than 20% as a result. Only a relatively small amount of the multinational firm’s business is conducted in the EU, but on top of the adverse effect that might have, any hit to the pensions and savings business would be expected to have knock-on effects in the UK too.

The question is whether we think the sell-off has been overdone. I think it has and that gives income seekers ready to take a bit of a risk on Legal & General’s chances of coming through the crisis in good shape an opportunity to bag some great dividends on the cheap.

The forecast yield for this year now stands as high as 7.7%, rising to 8.2% for 2017. There’s some safety in there too, as a dividend like that has room for a modest cut while still offering a great yield. And don’t forget, buying now would lock-in higher effective yields for life, not just for this year.

Pharma cash

We can find more big dividends at GlaxoSmithKline (LSE: GSK), another share whose business should be largely unaffected by EU fears. In fact, we’ve even seen a small rise in the past week, to around the 1,500p mark, presumably given a small boost by investors seeking safer stocks after the FTSE rout.

How do dividend yields of 5.3% forecast for this year and next sound? They sound good to me, as have Glaxo dividends all through the troubled period when the company was working hard on its development pipeline while earnings fell due to the key patents expiring and rising competition from generic alternatives.

That earnings slump is expected to end now, with EPS rises forecast for this year and next, and that makes GlaxoSmithKline look like another great long-term dividend pick to me.

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Alan Oscroft has no position in any shares mentioned. The Motley Fool UK owns shares of and has recommended GlaxoSmithKline. The Motley Fool UK has recommended Royal Dutch Shell. We Fools don't all hold the same opinions, but we all believe that considering a diverse range of insights makes us better investors.